Question

Prove by mathematical induction the following statement:

Answer 1

Answer:

In Exercises 1-15 use mathematical induction to establish the formula for n ≥ 1.

1. 1

2 + 22 + 32 + · · · + n

2 =

n(n + 1)(2n + 1)

6

Proof:

For n = 1, the statement reduces to 12 =

1 · 2 · 3

6

and is obviously true.

Assuming the statement is true for n = k:

1

2 + 22 + 32 + · · · + k

2 =

k(k + 1)(2k + 1)

6

, (1)

we will prove that the statement must be true for n = k + 1:

1

2 + 22 + 32 + · · · + (k + 1)2 =

(k + 1)(k + 2)(2k + 3)

6

. (2)

The left-hand side of (2) can be written as

1

2 + 22 + 32 + · · · + k

2 + (k + 1)2

.

In view of (1), this simplifies to: