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:
