   Chapter E, Problem 34E

Chapter
Section
Textbook Problem

Find the value of the sum.34. ∑ i = 1 n i ( i + 1 ) ( i + 2 )

To determine

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

Explanation

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

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