Mathematical Induction Math Example 3

Follow the full solution, then compare it with the other examples linked below.

medium

Prove by induction:

1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}

for all

n \ge 1

1
Base case ( $n=1$ ): LHS $= 1$ . RHS $= \frac{1 \cdot 2 \cdot 3}{6} = 1$ . True.
2
Assume true for $k$ : $\sum_{i=1}^{k} i^2 = \frac{k(k+1)(2k+1)}{6}$ .
3
For $k+1$ : $\sum_{i=1}^{k+1} i^2 = \frac{k(k+1)(2k+1)}{6} + (k+1)^2 = \frac{k(k+1)(2k+1) + 6(k+1)^2}{6}$ .
4
Factor out $(k+1)$ : $= \frac{(k+1)[k(2k+1) + 6(k+1)]}{6} = \frac{(k+1)(2k^2+7k+6)}{6} = \frac{(k+1)(k+2)(2k+3)}{6}$ .
5
This equals $\frac{(k+1)((k+1)+1)(2(k+1)+1)}{6}$ . QED.