BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter F, Problem 38E
To determine

To prove: The formula i=1ni3=[n(n+1)2]2 by mathematical induction.

Expert Solution

Explanation of Solution

Let Sn be the formula i=1ni3=[n(n+1)2]2.

Consider the formula i=1ni3=[n(n+1)2]2 for n=1.

i=11i3=13=[1(1+1)2]2=1

That is, S1 is true.

Assume that that the formula Sn is true for n=k.

That is, i=1ki3=[k(k+1)2]2.

To show the formula is true for n=k+1.

i=1k+1i3=i=1ki3+(k+1)3=[k(k+1)2]2+(k+1)3=(k+1)24[k2+4(k+1)]=(k+1)24(k+2)2

Further simplified as,

i=1k+1i3=[(k+1)(k+2)2]2=((k+1)[(k+1)+1]2)2

Thus, the formula is true for n=k+1.

Therefore, the formula Sn is true for all values of n by mathematical induction.

Hence the formula is proved.

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