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."