1+3+5+...+(2n+1)

Quandaries and Queries

Prove that 1+3+5+...+(2n+1)= (n+1)² for all n greater than or equal to 1.

Hi Emma,

Suppose that we use S to designate this sum, that is

S = 1 + 3 + 5 + ... + (2n+1)

There is a nice way to evaluate S that starts with evaluating 2S by writing the sum forwards and and then backwards.

2S = 1 + 3 + 5 + ... + 2n-3 + 2n-1 + 2n+1 +

2n+1 + 2n-1 + 2n-3 + ... + 5 + 3 + 1

Now add the terms in the sum by first adding down.

2S = 2n+2 + 2n+2 + 2n+2 + ... + 2n+2 + 2n+2 + 2n+2

= (2n+2) (the number of terms)

The terms in the sequence

1, 3, 5, ..., 2n-3, 2n-1, 2n+1

are half the terms in the sequence

1, 2, 3, 4, ..., 2n, 2n+1, 2n+2

and thus the number of terms in the sequence

1, 3, 5, ..., 2n-3, 2n-1, 2n+1

is ²ⁿ⁺²/₂ = n+1. Thus

2S = (2n+2) (n+1)

and hence

S = (n+1) (n+1) = (n+1)²

Penny