Question

What is the mathematical induction of
1+3+5+...+(2n-1)=n^2

Answer 1

Let
`S(n)` be the statement that


`1+3+5+\cdots+(2n-1)=\displaystyle\sum_(k=1)^n(2k-1)=n^2`

Check to see that
`S(n)` holds for
`n=1`:


`\displaystyle\sum_(k=1)^1(2k-1)=1=1^2\implies S(1)\text{ is true}`

Now suppose that
`S(N)` is true, and use that assumption to show that
`S(N+1)` must also be true. When
`n=N+1`, you have


`\displaystyle\sum_(k=1)^(N+1)(2k-1)=\sum_(k=1)^N(2k-1)+2(N+1)-1`

`=N^2+2(N+1)-1`

`=N^2+2N+1`

`=(N+1)^2`

and so
`S(N+1)` is also true, thus proving the statement is true for all
`n`.