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This is my account of going to see Morning Musume。 at Anime Expo in Los Angeles last week, which is a bit delayed due to technical and other difficulties. It was quite an adventure, and there is much more to say than I have room for here, but this should be a nice recap of my experience. I’ve left out parts that I think have been covered comprehensively elsewhere, like details of the MM Q&A panel and the concert. So, without further ado, LET’S STARTING!!! (o_0)

**Preparation**

When it was announced that Morning Musume。 would be signing autographs, I wanted to make the experience super awesome, so I got some additional merchandise that hopefully I’d be able to get signed while at AX. This included a full-size *Papancake* poster (Taiwan version):

(The ~~Hello!~~ Hello Kitty toaster is my roomie’s, not mine. ^_^)

This poster hung above our breakfast/dinner table for a while, and it was definitely enjoyable to eat breakfast with Koha (or eat breakfast while being stared at by Koha…).

I also got a *Best☆Kirari* poster and a Koha mini-poster, and my roomie got a *Kirari to Fuyu* poster.

After it was announced that we could get a different item signed by each member (for up to nine separate items), I wanted to do just that, since having all of their autographs on one item is kinda boring. Things didn’t exactly turn out as planned, but I was going to have these items signed:

*Mini Moni ja Movie: Okashi na Daibouken!*DVD cover — Takahashi*Souda! We’re ALIVE*CD cover — Niigaki*Hare Ame Nochi Suki ♡*CD cover — Kamei*Ai no Sono ~Touch My Heart!~*CD cover — Michishige*FIRST KISS*CD cover — Tanaka*Papancake*poster — Kusumi*Onna ni Sachi Are*CD cover — Mitsui*Shouganai Yume Oibito*CD cover — Junjun*Platinum 9 DISC*CD cover — Linlin*Osaka Koi no Uta*CD cover — HANGRY

If it turned out I’d be able to attend more than one autograph session, I’d get something else signed by everyone. Or maybe everyone except Koha, because I had a ton of Koha stuff. ^_^

In addition, since giving gifts was allowed, I decided to give each member one. Being broke, I decided to get MIT t-shirts for Aichan and Sayumi and a plush lobster for Koha. On top of that, each member would get an individual personalized card.

I had a lot to say to Koha, Aichan, and Sayu (who also happen to be my roomie’s favorite members—we like a lot of the same things; it’s like we’re the same person ^_^), so I filled up their cards, but I didn’t know what to say to the others. So rather than putting something generic on all the remaining cards, I came up with some pretty random messages for a few of the other members.

Eri got a pretty terrible (and self-referential) haiku:

亀が好き | kame ga suki | I like turtles |

亀井 大好き | Kamei daisuki | I like Kamei a lot |

俳句です | haiku desu | this is a haiku |

It was so bad it deserved another haiku to comment on its atrociousness:

詩は酷い | shi wa hidoi | poetry so bad |

墓に芭蕉の | haka ni Bashou no | Bashou in his grave |

寝返りね | negaeri ne | is rolling, isn’t he? |

Continuing with the haiku theme, I decided to write Reina some nonsensical Dadaist ones, all having a seasonal theme but otherwise not making a whole lot of sense (though they all have some extratextual resonance of one kind or another) … these actually took a while to write; nonsense isn’t as easy to write as it might seem.

夏休み | natsuyasumi | summer vacation |

サスカチュワンに | Sasukachuwan ni | to Saskatchewan |

果実 飛ぶ | kajitsu tobu | fruit flies |

紅葉の絵 | momiji no e | picture of autumn colors |

手押し車は | teoshi kuruma wa | the wheelbarrow |

青いです | aoi desu | is blue |

冬の道 | fuyu no michi | winter road |

あいにくケーキ | ainiku keeki | unfortunately the cake |

うそじゃない | uso ja nai | is not a lie |

春か猫 | haru ka neko | spring or cat? |

量子力学 | ryuushirikigaku | quantum mechanics |

知らないよ | shiranai yo | I don’t understand |

Having run out of creative haikujuices, I offered Aika the Y combinator in PostScript:

Apologies to Risa, Junjun, and Linlin … yours weren’t as spectacular, alas, other than a banana drawing for JunJun. :-o

As for the other three, Aichan got a green MIT t-shirt with a design like this but in black and white:

Her card had this on the back (“MIT ♡ Aichan”):

Inside, I told her how I really liked *Q.E.D. Shomei Shuryo* especially due to its MIT connections and that I knew of at least five current MIT students coming to AX just to see Morning Musume。 (which is a considerable number given that the school only has about 10,000 students), and several more who are H!P fans.

Due to her professed interest in math, Sayu also got an MIT t-shirt, but one that was more disguised. It had this design (not my photo, and not the shirt I bought):

In her card, I commented that the three letters clued by the formulae also happened to be the first three letters of her family name in Kunreishiki romanization (*Mitisige*). It’s like Sayu and MIT were made for each other… XD SayuMIT!

The back of Sayu’s card (featuring Euler’s identity):

Koha’s was by far the longest, as I had plenty to say. I told her about how her music was largely responsible for getting me into the H!P fandom, and that the “sine cosine tangent” line from “Hana wo Pu~n” was too awesome. I drew diagrams corresponding to her hand movements in the PV (the angle is in the bottom left, and she outlines the two sides whose ratio is the sine, cosine, or tangent of the angle; note that in Japanese convention, you put the denominator in a ratio first—1/3 is “sanbun no ichi”):

(The above was actually my first ever YouTube video, posted under my old account, back before this blog even existed.)

Also, I told Koha I shared this clip in my electronic music class and passed around my copy of *Kirarin☆Land*, and that my classmates and the instructor were very impressed (which is true … one classmate even wanted a rip of the album tracks).

In addition to this, I wrote down all eight of the double dactyls I had written to celebrate Koha’s birthday last year on a larger card and rolled it up.

That sums up my preparation, I think. Now to attend my first ever non-classical concert and my first ever anime convention! XD

**Day 0 – Airport Awesomeness and Line-waiting Lameness**

I flew into Orange County on June 29, and stayed there with my relatives for a week, other than the days of the convention. As this is irrelevant, I’ll jump to Day 0, July 1, when Morning Musume。 arrived at the airport.

I wasn’t originally planning on going to meet them at the airport, due to my lack of transportation and not wanting to inconvenience my relatives. But as June 30 rolled by without a sign of the group, it became clear that they’d definitely arrive on July 1, which was eventually confirmed by a fellow passenger on their flight.

I decided to get myself to the airport by public transportation only, as part of the charm of visiting a new city for me is getting lost in its public transportation, which inevitably happens regardless of how much planning I do. This has happened to me with Boston, New York, San Francisco, and Chicago, so I was eager to add Los Angeles to the list.

It took me 3.5 hours to get to LAX, using two local Orange County buses, an Orange County express bus, two Metra trains, and an airport shuttle. I did in fact get lost, as I failed to request the Harbor Freeway station stop to switch to the Metra Green Line. The bus didn’t announce its stops, and since no one requested that stop, it drove right on by. :-o Eventually, we got to downtown LA, driving right by the convention center, so that was my first glimpse of the place I’d be spending most of the next few days. I got off at 7th street and took the Metra from there, adding about forty-five minutes to my planned trip.

Arriving at the airport with plenty of time to spare, I saw that a small group had already gathered at the waiting area in the Tom Bradley International Terminal:

It was here that I met Bryan and Jen, hosts of the truly epic Hello! Party to come, and Claudia and Ivet, creators of the 3,2,1 BREAKIN’ OUT! OPV that won the Morning Musume。 prize. XDDD

After much waiting, during which several passengers from another flight from Japan did a double take after seeing the huge banner we had up (“Ehh~~??? Morning Musume。‽‽‽”), the members themselves appeared, to much fanfare and chaotic screaming:

We chased after them (from a respectable distance) and saw them get into a white van. Seeing them up close (and getting waves from them ^_^) was so awesome! One fellow passenger on their flight, after asking what all the commotion was and being told they were a popular J-pop group coming to America to perform for the first time, had this to say (regarding Aichan, I think): “She had nice legs.” :-o

I next went to the LA Convention Center with Bryan, Jen, Claudia, and Ivet, and got in line for badges. I was melting and hungry, so I parted ways and went to grab food, which in retrospect was probably not the best decision. Returning to the line a bit later, I saw that it had grown to a ridiculous size. Oh well. Unfortunately they had computer trouble, so I ended up waiting 90 minutes in the blazing sun. Ugh. And after that line came the concert ticket line, which was fortunately mostly in the shade. After four hours of waiting, I finally had badges and concert tickets for myself, my roomie Kyttyee, and our dear friend Aaron the Apathetic Wota (whom I will affectionately call A~ron in this post), who earlier this year had managed to accomplish the amazing feat of unintentionally attending a Johnny’s concert in Japan.

I also met up with CatchFiveBats, maiZe, johpan, Greg, and Paul Thomas.

Unfortunately, this meant I missed the last express bus out of LA, so I had to get a ride from my relatives. :-/

**Day 1 – Opening Ceremony, Q&A Panel, and More Waiting in Line**

The next day, A~ron and Kyttyee picked me up, and we arrived at the convention center in time for the opening ceremony, details of which you can find elsewhere. Afterward, A~ron and I got in line for the exhibit hall, which was ridiculously long and extended the entire length of the South Hall and a bit further. Once we got inside, shortly after noon, we rushed to the JapanFiles/JPopHouse booths to get autograph tickets. Unfortunately, the lines for the two booths had merged, and the staff were trying to untangle them. As it turned out, I was in the JPopHouse line when I thought I was in the JapanFiles line, and the rearranging resulted in the section of the line I was in getting shuffled to the back of the JapanFiles line. So after thirty minutes of waiting, I was now back at the end of the line. >_<

A~ron, having gotten into what looked like the JPopHouse line, ended up further ahead of me in the JapanFiles line, and was able to get a HANGRY ticket, but no MM ticket. So we managed to get something after all. Later on, we were able to get a second HANGRY ticket from JPopHouse after the panel, as they still had them for a while longer.

We headed to the panel and got there as the one before was letting out. We were able to get seats at the right of the fourth row, and after the panel was over (details of which you can find elsewhere), we got our autograph tickets and followed the directions to the back of Hall K in the Exhibit Hall. There we found hundreds of people running around in circles following each other, quite an amazing thing to experience. After overhearing some more sensible directions from the info booth, we headed upstairs to Room 301 and found the actual autograph line. Unfortunately, there were two lines on opposite sides of the building, and we were told to go to the other line, despite being right by one line of people who had already gathered. I decided to go to the other line while A~ron secured a spot for Kyttyee in the first line. The other line, when I got to it, wasn't much longer, so I stayed there. As it turned out, that was a mistake, as Kyttyee was able to get autographs in like 30 minutes (and a 3-minute conversation with Aika! :-o), while I ended up having the door shut in front of me with about twenty people ahead of me (I was #182).

Oh wells. Try again on Saturday?

We (CFB, Greg, maiZe, joh, and Paul, I think) headed down to the JPopHouse booth and got in line (yet again!) for some H!P merchandise. I bought a photoset and a *Platinum 9 Disco* poster, since the autograph session had specified one item for all members to sign, and I didn’t have anything I wanted to have signed that had all nine members on it. I also purchased some glowsticks and glow bracelets from JapanFiles. They had run out of all the colored glowsticks, but luckily found they still had a box of the red ones (Koha!!) as I got there. So yayz.

Having not eaten anything since dinner the day before, I was ready for some dinner, which I had at IHOP with CFB. Afterward I chilled for a bit with CFB and friends. We heard that Morning Musume。 was staying at the Westin, and discussed going there, but alas, I never actually went there the whole time I was in LA. :(

Returning to the convention center, exhausted, I napped in the lobby (should have brought a save point to set up next to me XD), and eventually got to the Econolodge motel in Hollywood or somewhere, where I would stay the next two nights with Kyttyee and Sekai no Melody friends Vivi and Kimi. Before catching some Z’s, we sang, danced, and listened to some of my mashups, and fangirled over the Koha posters Kyttyee and I had brought. ^_^ Kohaaaaa~~~!! **squee!**

**Day 2 – Concert, HANGRY, and Hello! Party**

In the morning, Kyttyee, A~ron, and I found ourselves back at the convention center for some more awesomeness. We met up with our friend “Edwin », who introduced me to Morning Musume。 in the first place, back when I was just getting into J-pop way back in 2007. Unfortunately, his choice of “LOVE Machine” left me disappointed with the group (I stopped watching the PV a third of the way in because I couldn’t stand it), and it wasn’t until several months later that I found that MM had awesome songs I liked.

Anyway, “Edwin » (being “Edwin ») didn’t stick around for the concert (!) and parted ways with us. We made our way over to the main hall, and somewhere along the way, I lost the poster that Kyttyee, Vivi, and I had made for Aichan (saying “MIT ♥ Aichan! We’ll always support you! Ganbatte ne!”) and took pictures of us with in front of MIT’s Great Dome, which we were planning to give to Aichan as a present (sorry! :( ). In addition, I had forgotten my glowsticks and glow bracelets at the hotel, so all I had were the eleven glow bracelets that had broken and which I was wearing. Kyttyee got us some more glowsticks (no red, alas…) and glow necklaces, and while waiting for the concert to start, I hooked them together into an awesome glowstarball that I waved during the concert (though it would have been even more awesome with non-dead glow bracelets forming the star, and with red glowsticks).

I must say, I was disappointed with “LOVE Machine” being performed, but the rest of the concert was pretty cool, other than having “3, 2, 1 BREAKIN’ OUT!” performed way too many times and being too short overall. But the medley and “Sono Bamen de Bibiccha Ikenai jan!” were definitely awesome (I use that word too much…).

Afterward, Kyttyee and I got in line for the HANGRY autograph signing, and since they wouldn’t let us form a line past a certain point, we walked around in circles and grabbed some food, which we ended up eating in line and actually brought into the autograph booth with us (“Hi HANGRY! Have some nachos!”).

ZOMG, HANGRY in person! It was surreal. I gave her the card I had written for her, and asked her to sign *Osaka Koi no Uta*, which she did, as “HANGRY → Yossie” (in hiragana):

Due to the excitement (which for the second time I just typed as “exciteness” … ???? O_o), I think I forgot to shake her hand. Blargz.

I grabbed dinner at IHOP (the third time I ate there while in LA), and headed to the awesome Hello! Party, where I encountered lots of fellow H!P fans, including Amy, Rocky, Lysa, CK (who I had met earlier in the day), mp122984, and delrey28.

The party was super awesome, and I stayed until the end. Kyttyee and I were planning to karaoke Renai♥Chance!, but we were too far back in the queue and didn’t get to. But I’m still eager to see what mashup karaoke sounds like and what kind of reaction it would get from a large gathering of H!P fans. (Next Hello! Party? XDDD)

**Day 3 – Ch-ch-chance!**

Waking up at 8:00 was FAIL. Checking online, I found that the line for the morning autograph session was already more than a hundred people long. Nevertheless, we rushed to the convention center and found the line (thereby missing the second half of the TSUNKU♂ panel >_<). Kyttyee and Kimi chilled in the standby line, while I joined the ticketed line with my ticket from Thursday. And to my surprise, the line was shorter than I expected, with me being only #183 in line (one spot higher than on Thursday). Since Thursday's session had started late and ended early and was rather unorganized, I was hoping it'd be speedier this time and get through more than 160 people.

Unfortunately, after waiting 90 minutes and having had my hopes up, I had the door shut in front of me again, with only twelve people in front of me. >_<

Thrice thwarted!

Dejected, I made my way down to the exhibit hall and joined Kyttyee and Vivi for some lunch before going over to the afternoon autograph session in hopes of finding someone who was getting individual items signed and didn't care much for Koha (so I could get my *Papancake* poster signed instead). As it turned out, they had changed their policy at the last minute to be only one item for all the members (like the other sessions), so this wouldn’t have worked anyway. But at the time I didn’t know that, which worked out in my favor, since after unsuccessfully asking a few people in line, I ran into Claudia and Ivet, who had an extra ticket (thank you so much! you are awesome! XDDDD). Unbelievable joy!

So I got in line (in front of Amy and Mozenator) and had all my individual items out, along with everyone’s gifts. It was hard to hold everything at the same time and I accidentally dropped some of my stuff on the floor (>_<). The line was moving really quickly, compared to the previous lines I was in, and in no time, we were right in front of the booth MM were in, which I had walked past before getting in line, seeing them through the curtains.

At this point, they told us that we could only have one item for all the members, so I was desperately trying to stuff all my individual items into my bag, pull out and unwrap my group poster, and keep from dropping everything on the floor at the same time while walking into the booth—where I found MM in *not their expected order* (!), so I had to re-sort my gift cards as well. I handed my poster to a staff member (who promptly unrolled it in front of Reina, sitting at the front) and asked if it was ok to give gifts, then handed over my pile of gifts as well. As a result of this chaos, I didn’t get to say anything to Reina other than a quick “arigatou gozaimasu” since she had already finished signing. >__<

Having failed to shake HANGRY’s hand on Friday, I wanted to shake hands with Morning Musume。 this time. Apparently this session was a “no handshakes, no touching, no chitchat” session, unlike the previous two (probably because they needed to rush through 200 autographs—I think only 200 of the announced 300 tickets were actually given out) in a short amount of time. But I didn’t know that—I assumed handshakes were ok because they were fine in the previous sessions. Either they didn’t announce the policy for this session, or I was too busy trying to do too many things at once and didn’t register what they were saying. So I came in expecting to be able to shake hands and say a few reasonably short things to the MM members.

I remembered to ask for a handshake only after Reina was already done signing, so I attempted to shake Linlin’s hand. She didn’t appear to notice (and my brain was too frozen to say anything), so I retracted my hand and moved on. >_< I don't think any of the staff noticed either, because I didn't get any response from them.

Next was Aichan. I managed to say hello to her and tell her I'm an MIT student (showing her my MIT ID card), and that I loved her drama *Q.E.D. Shomei Shuryo*. She seemed pretty excited and said “arigatou”. ^_^ I was cut off by a staff member saying something (presumably telling me to hurry up), so I thanked her and moved on. XDDD

After Aichan were Risa, Aika, and Eri. Unfortunately, I don’t remember what happened at this point because I got an unexpected “HI!!!!” and enthusiastic waves from both Eri and Aika (at different points in time) while I was still in front of another member. I hope I responded appropriately to their greetings and remembered to thank all three of them for their autographs… :o

Next was Sayu, and I was going to say something mathy, but everything was rushed and confused, so I wasn’t able to say much. I do remember that my poster rolled itself up, and I got a “sorry, sorry!” from Sayu and maybe Koha while I helped them unroll the poster.

Seeing Koha unoccupied (and looking pretty tired, I think), I greeted her with a “KONNICHI PA!!!” and got a “KONNICHI PA!!!” in return, along with an ever-delightful Koha-esque **8D** of utter glee (XDDDD). Since I had missed my chance to shake hands with all of the previous members, I offered my hand, and Koha, after a brief moment of hesitation (well, it *was* a “no handshakes” autograph session after all …), shook it (♥ ^_^ ♥). KOHANDSHAKE!!! XDDD **squee~!**

Unfortunately, I didn’t get to tell her how awesome she was, but hopefully she’ll have a chance to read the card I wrote for her.

I hope I didn’t completely skip over Sayu while doing this (if so, I’m sorry!). D-:

And I don’t even remember what happened with Junjun. O_o

And … curses! I was going to wish Sayu and Koha a happy birthday, and I forgot! >_< gahhhhhhhh … well, I put it in their cards, so I hope they saw them… ^_^

Overall, getting to see Morning Musume。 in person and interact with them was a blast, even if the session was extremely rushed, and the members didn't seem to be enjoying it too much (from other people's reports, apparently several members were told to hurry up and not to interact with the fans, and at least one fan had their hand slapped by a staff member for wanting to get a handshake >_<). I don't recall making eye contact with Reina, Linlin, Risa, Sayu, or Junjun (T_T), but being greeted by Eri and Aika and getting to talk a bit with Aichan and Koha (with a Kohandshake!) was definitely too awesome for words. XDDD

Afterward, there was much ecstatic squeeing as we met up with CFB, CK, joh, maiZe, Rocky, and Paul and looked at what we’d had signed. ^_^

And that is where I’ll end my report, because I left the con later in the evening and didn’t do anything on Sunday. Maybe I should have gone to the airport to see MM off (Koha was wearing glasses and Sayu, Aichan, and Junjun were wearing what appear to be snowboots! I totally approve, having worn glasses and snowboots at the con myself… XDDD), but it was nice just to relax after several days of rushing around, waiting around, and not getting enough sleep.

I’d like to thank Morning Musume。, HANGRY, and TSUNKU♂ for taking the time and effort to come to LA and meet us fans, and the AX/UFA/JapanFiles/JPopHouse staff who made all this possible, even if things could have been done a bit differently. And a bucketful of huge thanks to Bryan and Jen for organizing Hello! Party, and thank you to all H!P fans for your support, whether or not you attended AX; without you, this event wouldn’t have happened. Until next time! ^_^

I’ve been anticipating the new detective drama *Q.E.D.* starring Takahashi Ai and Nakamura Aoi since it looked like it would involve a lot of awesome math, and now that the first episode is out, we can see that there is indeed a lot of awesome math! And correct, non-trivial math to boot!

Hello! Project member + stereotypical MIT nerd = WIN!

Meet Touma Sou (Nakamura), stereotypical MIT math nerd! Who, having graduated from MIT at a very early age, is for some reason reattending high school … I’m sure this will be explained in future episodes …

Mizuhara Kana (Takahashi) does an impression of MIT! D-:

Of course I am highly amused that “Massachusetts Institute of Technology” appears on the screen at the same time as Takahashi, who should definitely drop by for a visit sometime. I can show you around this place, Aichan, if you’re ever in the area.

Since “Q.E.D.” is traditionally written at the end of a math proof, this first episode of course features a math proof. As the show’s primary purpose is entertainment and not educational, it doesn’t focus on the details, but the details nevertheless are there and they’re intellectually stimulating, so I’ll cover that aspect here.

To begin with, the teacher presents the following proof in class:

[If this is hard to read, it and the below proofs are available as a PDF.]

Let’s go through the details. The statement being proved is given in line (1). This is giving a closed-form expression for the sum of the product of three consecutive integers over a range. In line (2), the summed expression is expanded as a cubic polynomial: . This lets us use linearity of summation to split the sum into three parts (3). Then using summation identities, each of the three sums is expressed in closed form (4). In line (5), we factor out , and in line (6), we group like terms of the remaining expression to produce a quadratic polynomial: . This polynomial can be factored into , leaving the desired expression (7).

Of course, for Sou this is not nearly intellectually challenging enough. (The subtitles say that the teacher’s derivation is erroneous, but it’s correct as far as I can tell; if this is what Sou actually says, maybe the producers just wanted to avoid getting into details about the distinction between generalizing a theorem and offering a better proof of the same theorem.) Sou realizes that this fact can be generalized, establishing a more powerful theorem. Instead of having a product of just three consecutive integers , why not have arbitrarily many? Sou generalizes the theorem to apply to a product of consecutive integers: . And indeed, there is a closed-form expression for this as well:

Note that this reduces to the previous theorem if you substitute .

It’s a fun exercise to try to prove this theorem, which I encourage you to do if you’re interested. I did so, and I’ll present my own proof later on. As it turns out, Sou’s proof is simpler than mine, but due to camera angles and low resolution, it’s pretty hard to actually see what the proof is.

After much squinting and deducing, I finally figured it out:

This is pretty short and makes some jumps that may not be intuitive, so I’ll go through the details:

In line (9), each term of the summation is multiplied by a ratio equal to 1, which does not change the value of the expression. You can easily do the math and see that the numerator is equal to . In line (10), the denominator of this ratio is pulled out of the summation, since it’s a constant factor, and the two terms in the numerator are multiplied with the product , using the distributive law. Line (11) is probably the biggest jump in this proof; it’s using a telescoping sum, reducing a sum of terms by canceling out intermediate pairs. If you plug in for , you get , and likewise, for , you get Notice that the terms cancel when you add these together. If you keep doing this all the way to , all the intermediate pairs cancel out, and you’re left with just two terms: . The second term vanishes, so the overall expression is reduced to (11), which completes the proof.

The following is my own proof, which is a bit lengthier and uses binomial coefficients, along with Pascal’s second identity. Let’s first prove Pascal’s second identity, stated here as a lemma:

We use the principle of induction, which is showing that a statement holds for some base case (here ) and showing that if the statement holds for some , then it has to hold for as well. Putting these parts together, you prove that the statement holds for all integers greater than or equal to the base case.

Lines (13) through (17) prove the base case. This makes use of the definition of the binomial coefficient , which for is equal to 1.

The inductive step is proved in lines (18) through (25). We assume the statement holds for and show that it also holds for . In line (18), we split the sum of terms into two parts. The first part is identical to the sum of terms, which we can replace with by the inductive assumption (19). In line (20), we replace the binomial coefficients with their equivalent factorial ratios, and in line (21), we factor and out of the two denominators, respectively. In lines (22) and (23), we factor out common factors of both terms, and add together the remaining fractions. Combining the two parts, we get (24), which can be written as (25), thus proving the lemma.

My alternate proof is as follows:

In line (26), we rewrite the product of consecutive integers as a ratio of factorials. We then reindex the summation (27) for convenience, so that starts at 0. Next we multiply the entire expression by and divide it by the same amount (28). Each term of the summation can be expressed in binomial form (29). Using Pascal’s second identity, we can rewrite the sum as (30). Turning this back into a ratio of factorials, we get (31). We can cancel out a factor, leaving (32). Finally, expressing this ratio of factorials as a product of consecutive integers, we have (33), as desired.

(I’d stick a “Q.E.D.” at the end of that proof, but no one uses that anymore. The square box is sufficient.)

I’m definitely looking forward to more math and awesomeness in the upcoming episodes….

[The above proofs are available as a PDF.]

**EDIT:** Upon rewatching the episode, I discovered that Sou has excellent taste in reading material:

Douglas R. Hofstadter’s *Gödel, Escher, Bach: an Eternal Golden Braid*. I approve.

**EDIT 2:** Agggh!! What was I thinking? There’s a really easy inductive proof that I would have seen had I inducted on the right variable the first time. >_<

It’s, uh, self-explanatory.

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