# The value of the sum ∑ i = 1 n i ( i + 1 ) ( 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 34E
To determine

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

Expert Solution

The value of the sum i=1ni(i+1)(i+2) is n(n+1)(n+2)(n+3)4.

### 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=1ni(i+1)(i+2) expressed as follows.

i=1ni(i+1)(i+2)=i=1ni(i2+3i+2)=i=1n(i3+3i2+2i)=i=1ni3+3i=1ni2+2i=1ni=(n(n+1)2)2+3(n(n+1)(2n+1)6)+2(n(n+1)2)

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

i=1ni(i+1)(i+2)=n(n+1)[n(n+1)4+2n+12+1]=n(n+1)4(n(n+1)+4n+2+4)=n(n+1)4(n2+5n+6)=n(n+1)(n+2)(n+3)4

Thus value of the sum i=1ni(i+1)(i+2) is n(n+1)(n+2)(n+3)4.

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