   Chapter 13.1, Problem 25E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# In Problems 20-25, use the sum formulas I-V to express each of the following without the summation symbol. In Problems 20-23, find the numerical value. 25.   ∑ i = 1 n ( 1 − 2 i n + i 2 n 2 ) ( 3 n )

To determine

To calculate: The expression of i=1n(12in+i2n2)(3n) without the summation symbol.

Explanation

Given Information:

The provided sum is i=1n(12in+i2n2)(3n).

Formula used:

The value of the sum k=1n(xk±yk)=k=1nxk±k=1nyk.

The value of the sum k=1nk=n(n+1)2.

The value of the sum k=1n1=n.

The value of the sum k=1ncxk=ck=1nxk.

The value of the sum k=1nk2=n(n+1)(2n+1)6.

Calculation:

Consider the sum i=1n(12in+i2n2)(3n).

This can be written as,

i=1n(12in+i2n2)(3n)=i=1n(3n3n2in+3ni2n2)=i=1n(3n6in2+3i2n3)

Recall that the value of the sum k=1n(xk±yk)=k=1nxk±k=1nyk.

Thus,

i=1n(12in+i2n2)(3n)=i=1n(3n6in2+3i2n3)=i=1n3ni=1n(6in2)+i=1n(3i2n3)

Recall that the value of the sum k=1ncxk=ck=1nxk.

Thus,

i=1n(12in+i2n2)(3n)=i=1n3ni=1n(6in2)+i=1n(3i2n3)=3ni=1n16n2i=1ni+3n3i=1ni2

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