For each positive integer n, let P(n) be the formula 12 + 22 ++ n2 = n(n + 1)(2n + 1) 6

Chapter 5, Problem 3

(choose chapter or problem)

For each positive integer n, let P(n) be the formula

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

a. Write P(1). Is P(1) true?

b. Write P(k).

c. Write P(k + 1).

d. In a proof by mathematical induction that the formula holds for all integers n ≥ 1, what must be shown in the inductive step?

Text Transcription:

1^2 + 2^2 + cdots + n^2 = n(n+1)(2 n+1) / 6

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