# The value of the sum ∑ i = 1 n ( i 2 + 3 i + 4 ) . ### 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 31E
To determine

## To find: The value of the sum ∑i=1n(i2+3i+4).

Expert Solution

The value of the sum i=1n(i2+3i+4) is n3(n2+6n+17).

### 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(i2+3i+4) expressed as follows.

i=1n(i2+3i+4)=i=1ni2+3i=1ni+i=1n4=(n(n+1)(2n+1)6)+3(n(n+1)2)+4n=16[(2n3+3n2+n)+(9n2+9n)+24n]=16(2n3+12n2+34n)=n3(n2+6n+17)

Thus value of the sum i=1n(i2+3i+4) is n3(n2+6n+17).

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