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?