Baka to Tesuto to Syokanju:Volume7 The Fourth Question
Use mathematical induction [1] to prove the following:
1 + 3 + 5 + … + (2n-1)=n2 – (1)
- When n is a natural number.
Himeji Mizuki's answer
1. When n=1, equation (1) is calculated as,
- Left Hand Side = 1
- Right Hand Side = 1
2. Assuming that n=k is valid,
- 1 + 3 + 5 + … + (2k-1)=k2 - (2)
When n=k+1, the left hand side of (1) is written as
- 1 + 3 + 5 + … + (2k-1)+(2k+1)
- =k2 + (2k+1) (Obtained from equation (2))
- =(k+1)2
In other words,
- 1 + 3 + 5 + … + (2k-1)+(2k+1)
- =(k+1)2
(1) is valid when n=k+1.
From both equations (1) and (2), equation (1) is valid for all values on n.
Teacher's comment
That's correct. In mathematical induction, you need to prove that n=1 is valid. Also, when assuming that n=k is valid, you have to deduce that n=k+1 is valid. In this question, the key is that n is a natural number. On a side note, people often forget to put that n=1, so do take note when answering.
Tsuchiya Kouta's answer
"I'll prove here that (1) is correct - Tsuchiya Kouta"
Teacher's comment
Even if you try to sneak through by writing it in a style of a thesis, it's usless. The question mentioned that you have to prove it through 'mathematical induction', so while assuming that n=k, please write the equation that n=k+1 is valid.
Yoshii Akihisa's answer
"My judgment can hold."
Teacher's comment
"Please use an assumption."
“Oh! Akihisa, you're back.”
“Been tough on you, Akihisa.”
“...Welcome back.”
“Ah, mn, I'm back...”
As I finish the replenishment tests and return to the middle of the sports arena, Hideyoshi and the rest kindly receive my arrival. Seeing them like this at least alleviates the feeling of behind left behind a bit. But whether we won or lost, I still wished to see it through with everyone...
Thinking about this, my eyes turn to the class F rest area that's bounded by rope. At this moment, I realize that the guys in class are surrounding a strange box and arguing. What are they doing?
“PLEASE LET ME BE THE LUCKY GUY!”
“STOP YAPPING AND START DRAWING! THERE ARE PEOPLE BEHIND!”
“I GOT IT, I GOT IT! WHY ARE YOU GUYS SO ANXIOUS! ALRIGHT, IT'S THIS ONE—DAMN IIIIIITTTTT!!!”
“““NICE! CRY YOUR WAY HOME!!!”””
“Eh, what are they doing?”
I asked Yuuji who's beside me. Several classmates of mine seem to be dejected and unhappy. Why is that?
“Hm? You're talking about them? Just doing a lucky draw.”
“I know that. I just want to know why they're ding it.”
“Oh. The next event is the 2-people-3-legged race. The draw's to decide who goes with who.”
“Heh~so that's how it is.”
It's true that the choosing of partners in the 2-people-3-legged race is crucial. More than one's own ability, the most important thing is for both sides to be able to co-ordinate together.
“What's wrong, Akihisa? You're pretty calm.”
Hideyoshi asked me with a puzzled look, but there's no reason for me not to remain calm...
“Because it doesn't matter who's the partner. Guys and girls will be separated anyway--”
“This is a mixed event.”
“No problems at all. SUMMON!”
GO, MY SUMMONED BEAST! KILL THOSE GUYS FOR ME!!
“CALM DOWN, YOU IDIOT! DO YOU THINK YOU CAN SUMMON WITHOUT A TEACHER'S PERMISSION?”
“I CAN'T CALM DOWN AT ALLLL!! WHO IS IT? WHO'S THAT LUCKY GUY WHO GOT THE LUCKY NUMBER?”
“Relax, it's not decided yet.”
“Eh? Really?”
“...If that were the case, we wouldn't be quarrelling like this.”
“Ah, you're right.”
Lucky...the key to winning the 2-people-3-legged race is to not get anxious and match the steps of the partner. The bodies will have to be together. If there's any traitor who would do such a great thing to the girls, that guy will be massacred by the FFF.
“I see. That's why everyone's praying.”
“That's the case.”
Everyone's praying before they stand in front of the box, putting their hands together and praying.
“But, to think that Himeji-san and Minami would agree to play the 2-people-3-legged race with the guys as well. Yuuji, did you say something to them?”
Looking over there, the other girls seem to be paired up with the same gender. Himeji-san and Minami probably did that because there are only 3 people, Himeji-san, Minami and Hideyoshi and can't leave someone out?
Hearing my words, Yuuji gives a 'really, you...' look.
“What idiotic words are you saying? I would wish for guys and girls to be in different groups.”
“Eh? Why—oh ya, I understand. You'll be killed if something goes wrong.”
“Good that you can understand.”
Once Kirishima-san knows that Yuuji's partner is a girl, no matter how far Yuuji is, she would install some counter measures. There's no such thing as 'impossible' in a girl-in-love's dictionary—everyone knows that.
“More than that, I feel that the baseball tournament's more important.”
“...I agree.”
Yuuji and Muttsurini nod their heads. These people have already turned into Asure just to get their confiscated A-books. They would prefer the more practical honor of getting back their A-books 10,000 times more than getting honor and glory.
“If that's the case, the next match is decided, right?”
The middle (first) sports arena is used to hold the sports meet, and the second arena that's located behind the school compound was used to hold our second-year baseball tournament.. The gym is opened for the 3rd years. We're facing the 3rd years later...
“...They haven't finished yet. It's extra innings.”
“Oh, if we're successful, we get a walk over.”
Due to the time limit, this match will be extended to 7 innings. If they can't win, the match will be decided as a tie. It sounds nice, but both sides can't progress on. If that's really the case, we can take the next match without playing.
“Anyway, let's check the next match to game plan...let me see, the next match has maths, biology, modern literature, economics and geography.”
Yuuji took out the schedule to check. These are all basic, normal subjects.
“...No health education.”
Muttsurini sounded rather lonely when he said that as he glanced at the schedule in Yuuji's hand. Like what Muttsurini says, there's no health education for every match.
“If that's the case, do you want to take part in the sports meet event, Muttsurini?”
“You're right. We'll do that for the second round.”
“...I got it.”
To think that Muttsurini would actually leave the frontlines. However, since there's no health education in the next few matches, it can't be helped. Besides, Muttsurini's other subjects are worse than me.
“Then we should readjust the batting order and the defensive alignments. The first one is maths, so Shimada will be the 1st batter...we'll let Sugawa handle the second. That guy seems like he can really bunt.”
“Then what about me? I can't handle maths.”
“You can't handle maths 'too'. Besides, even if we leave you at the back few batters, your biology scores are atrocious, so I might as well just leave you at 3rd. It's good if you can really bunt. If you'rere successful, we should be able to get 1 point.”
“Mn, got it.”
Yuuji decided our batting order and defensive positioning.
Just when Yuuji and I are discussing--
“Well, erm, Akihisa-kun...”
A voice called me from behind. This voice...Himeji-san?
“Hm? Himeji-san, what's wrong?”
Turning around, I see that Himeji-san and Minami are standing behind me. They're looking really serious. What's going on?
“It's not really something big...”
“Well...Akihisa-kun, what's your number?”
“Eh? You're asking me for my number...”
Ah, should be the batting order, right? As expected of Himeji-san, being so serious when she does things. She would actually check the defensive position and batting order. Though she's not really good at understanding sports, it seems like he's seriously thinking about this match.
Uu, my batting order...
“I'm 3. Number three.”
““WAH!!””
After I answered that, Himeji-sana and Minami seem to whimper for some reason. Is my batting order weird?
“Uuu...I know the chances are slim, but I thought that it could be possible...”
“That's too much. This is too saddening...the one chance to stick on. This is a rare chance...”
Both of them are holding onto a little piece of paper each as they becried.
“AH~so you're talking about that. Don't worry, you two, Akihisa was just talking about the batting order. He hasn't drawn the number for the 2-persons-3-legged race.”
Yuuji said to said while sounding like he's lecturing them.
““Eh?””
Himeji-san and Minami, who were in a moment of melancholy just a second ago, lifted their heads up at the same time. Hm? Was the number they were talking about regarding the 2-persons-3-legged race?
“Oh ya. I forgot all about that. Got to take one number!”
Who can take it if those guys take the wonderful opportunity to be with the girls!?
“Ah, that...Akihisa-kun!”
“Wait, Aki!”
As I was about to go for the box of numbers, Himeji-san and Minami suddenly stopped me. Uu...if I don't hurry up, my happiness will be taken away.
“What? Is there something?”
“No. We...well...what should I say...I'm 7...”
“My number's 6...”
Himeji-san and Minami seem to say these numbers gingerly. 7 and 6? I see. So I'll strike jackpot if I get these numbers, right--
““(PLEASE) DON'T DRAW THIS NUMBER!””
Seems like they really hate me.
“I, I got it...then, I'll draw one number then...”
“I say, you two. Those words may have already caused Akihisa to misunderstand...”
“Be, because it's Akihisa-kun...if we say to him 'please draw this number'--”
“That, that's right! When that happens, it will become the opposite of what we wish for. For some reason, I keep feeling that he will get Sakamoto's number or something...”
“...Really, it's been tough on you two...”
Sigh, though I never had much luck, I can't possibly get a good draw anyway. But I'm really sad to be told them by them...I drag my heavy footsteps and arrive in front of Sugawa-san, who's watching over the box.
“Alright, Yoshii, draw your fate.”
“Uu, okay...”
With Sugawa-san staring at me mysterious, I reach my hand into the box. It'll be heaven if I can draw 6 or 7, so I have to do my best. Uu...okay, this one!
I immediately pull out the number. The number written on the paper is--
“Ah, 6...”
“KILL HIM!!!”
“““YES, HIEROPHANT!!!”””
“WHAT'S GOING ON? I GOT SURROUNDED!”
I haven't even said my number out loud, and a large group of people in black hoods surrounded me. I was careless! Saying our my number so openly like that, is it the end of me...
“6, 6...I see, so Aki will be with me...”
“Uu...Minami-chan looks so happy...”
“Do, do I? When do I look happy...”
“You liar! I can see your face glowing...”
“Uu...”
“I think Minami-chan will use this chance to touch Akihisa-kun's chest and butt...that's too sneaky...”
“Wh, what are you saying, Mizuki! How can I do such a thing—eh, what? What did you just say? Touching...Mizuki, what were you talking about...”
“Ah! No, nothing! I'm not talking about touching him sneakily. Th, that...I'm just talking about improving the relationship with Akihisa-kun. I just said it wrongly!”
“Mizuki..what were you planning to do to Aki...”
In a blink of an eye, I got wrestled down onto the floor by them as they forcefully take away that piece of paper in my hand. These guys...these techniques aren't moves that amateurs can do!
“Alright, this no.6 will be auctioned.”
“VERY GOOD, NO MATTER HOW MUCH YOU OFFER, MIHARU WILL BUY IT!”
“““WHY IN THE WORLD IS SHIMIZU-SAN HERE!?”””
“Sorry, but this is a class affair—hm?”
“What's wrong, leader?”
“No, this..isn't 6. It's 9. Yoshii just looked at it in the opposite way.”
Sugawa-san opens the number in his hand. The number 9 has an arrow to indicate how it should be read.
“What the hell, so it's 9. That shocked us...”
“Yoshii really loves to create commotions.”
“Causing us to spend so much effort on this.”
“It's all Yoshii's fault for being unable to see such a simple number like that. How tupid.”
The FFF members grumbled as they turn away. That was close. If I really got 6, I would have been killed...
“Well, Aki...”
While I'm stroking my joints, Minami walks up to me.
“Hm? Minami, what's up?”
“What should I say? Well...before the meet, why don't we tie our legs toward and practise a bit? We need to practise on chemistry, so I think we should practise to match our breathing and rhythem...”
Minami starts to warm up as she asked me.
Practise for the 2-persons-3-legged race? I see. So Minami's seeing that as really important and actually willing to make a contribution to the class like that. That's nice. In contrast, those classmates of mine who would execute me out of jealousy are really rotten until I want them to the mud in their nails.
“Ah...but I feel that it's best to train with your own partner, right? It's more effective to train together with your partner.”
“I'm not deliberately closing my distance with your body—eh? What did you just say?”
“I just got 9. Didn't you get 6, Minami?”
I open my paper I just drew to Minami for her to see.
That's a pity. If I could partner with the athletic Minami, I can perform better in the 2-persons-3-legged race. And besides, there would be a lot of happy things happening as well...
“Oh, I see. So it's like that...if that's the case, it doesn't matter...”
Hearing me say that, Minami shrugs her shoulders and looks lost as she walks away. Did she feel that it would be easier to pair up with me? We do work together well. If she really thinks this way, it would be really great.
“I'm sorry, Minami-chan. The truth is that, I heaved a sigh of relief just now.”
“Don't worry Mizuki...I think I would do the same thing if our roles were reversed...”
Minami walked to Himeji-san, and both of them said something. It seems to be something involving girls. I shouldn't eavesdrop, and besides, it's time to go back to Yuuji and the rest.
After keeping the number I opened properly, I turn around to return to my usual friends. Yuuji and the rest seem to have finished discussing about the baseball match as Yuuji passed me my batting order number once I got there.
“What's wrong, Akihisa? You didn't get the number you want?”
Yuuji mocked me. If I really got the number I wanted, I would have been buried, right? In a certain sense, maybe such an outcome is really the jackpot.
Speaking of which, who's my partner?
“Oh ya, Yuuji, what's your number?”
“Hm? Well, I haven't drawn yet. I wanted to draw once those guys calm down, but I forgot.”
“Me too.”
“...Same here.”
It seems like these people haven't decided on their partners in the 2-persons-3-legged race. Since the partner number 9 hasn't drawn his number yet, one of these three people will be my partner. If taht's the case...
“Please, Hideyoshi! You must draw 9!”
“Hm? What's wrong, Akihisa? You want to be partners with me?”
“Of course! I really want to be partners with Hideyoshi!”
If it's just Yuuji, Hideyoshi and Muttsurini, I would choose Hideyowhi without exception. Considering the honor of the class, it would be better to choose Yuuji or Muttsurini, but things aren't that simple. This world has things that are more important than winning.
“We should draw our numbers too. Besides, the class challenges is more important than the baseball tournament as per schedule.”
“...It will be bad if we don't take part in those events.”
“I feel the same.”
The 3 of them head to the box. As things go, only those 3 haven't drawn numbers yet. The only ones with an undecided partner are me, Himeji-san and Minami. In other words, once they draw the numbers, there will be tragedies of bloodshed.
While everyone's trying to hold back their saliva, Hideyoshi reaches into the box.
Please, Hideyoshi, get 9!
“Uu, I got 9--”
“BRING IT ON, YOU GUYS!! I'LL DEFEND THIS PIECE OF PAPER EVEN IF I DIE!!!”
“““DON'T THINK THAT YOU CAN LIVE TO SEE THE SUN TOMORROW, IDIOT!!!”””
Not willing to lose to the overwhelming killing intent, I shoued out and gripped my fists tightly. There's only a few minutes before the 2-persons-3-legged race begins. Until then, I must survive!
“--Sorry, saw it wrongly. It's 6.”
“““...”””
The killing intent immediately dissipated. At this same time, my enthusiasm disappeared. If it's 6, then Hideyoshi's partnered with Minami...I feel that it's a pity, but there's a sense of relief here...
If that's the case, there's only 7 and 9 left. The next one will either be my partner or Himeji-san.
“...(sssst).”
Muttsurini reaches into the box. To him, this is a battle of winning and losing, right? If he can be with Himeji-san, he can touch Himeji-san's amazingly self-aware parts. That guy will jump into the vortex of happiness even when he knows that he would die. I don't want him to get such an envious outcome!
Muttsurini finally pulls out that piece of paper that decides his fate and slowly opens it. Now, what will he get...
“...9--”
“FAREWELL!!”
“DON'T YOU DARE ESCAPE!! EVERYONE, CATCH SAKAMOTO AND OFFER HIM FOR A BLOOD SACRIFICE!!!”
“OSSSUU!!”
