   Chapter 13.CT, Problem 10CT ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

#### Solutions

Chapter
Section ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

# Use mathematical induction to prove that for all natural numbers n . 1 2 + 2 2 + 3 2 + ... + n 2 = n ( n + 1 ) ( 2 n + 1 ) 6

To determine

To prove:

For all natural numbers n, 12+22+32+...+n2=n(n+1)(2n+1)6.

Explanation

Given:

The number n is a natural number.

Approach:

Use the Principle of Mathematical induction.

Calculation:

Let P(n) denote the equality 12+22+32+...+n2=n(n+1)(2n+1)6.

For n=1, P(1) is true.

12=1(1+1)(21+1)6=66=1

Suppose for n=k, P(k) is true,

P(k)=12+22+32+...+k2=k(k+1)(2k+1)6 ……(1)

Let n=(k+1), then left hand side of P(k+1) is:

12+22+32+...+(k+1)2=12+22+32+

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