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 35E
To determine

To find: The value of the sum i=1n(i3i2).

Expert Solution

Answer to Problem 35E

The value of the sum i=1n(i3i2) is 14n(n3+2n2n10).

Explanation of Solution

Definition used:

If am,am+1,...,an are real numbers and m and n are integers such that mn, then i=mnai=am+am+1+am+2++an1+an.

Theorem used:

Let c be a constant and n be a positive integer. Then,

i=1nc=nc, i=1ni=n(n+1)2 and i=1ni2=n(n+1)(2n+1)6.

Calculation:

By the above definition, the sum i=1n(i3i2) expressed as follows.

i=1n(i3i2)=i=1ni3i=1nii=1n2=(n(n+1)2)2(n(n+1)2)2n=14n(n+1)[n(n+1)2]2n=14n(n+1)(n+2)(n1)2n

On further simplification the value of the sum i=1n(i3i2) becomes,

i=1n(i3i2)=14n[(n+1)(n+2)(n1)8]=14n[(n21)(n+2)8]=14n(n3+2n2n10)

Thus value of the sum i=1n(i3i2) is 14n(n3+2n2n10).

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