# The value of the sum ∑ i = 1 n ( 3 + 2 i ) 2 .

### Single Variable Calculus: Concepts...

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

### Single Variable Calculus: Concepts...

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

#### Solutions

Chapter F, Problem 32E
To determine

## To find: The value of the sum ∑i=1n(3+2i)2.

Expert Solution

The value of the sum i=1n(3+2i)2 is n3(4n2+24n+47).

### 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(3+2i)2 expressed as follows.

i=1n(3+2i)2=i=1n(9+12i+4i2)=i=1n9+12i=1ni+4i=1ni2=9n+12(n(n+1)2)+4(n(n+1)(2n+1)6)=9n+6n(n+1)+(2n(n+1)(2n+1)3)

On further simplification the value of the sum i=1n(3+2i)2 becomes,

i=1n(3+2i)2=27n+18n2+18n+4n3+6n2+2n3=13(4n3+24n2+47n)=n3(4n2+24n+47)

Thus value of the sum i=1n(3+2i)2 is n3(4n2+24n+47).

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