   Chapter 4.2, Problem 26E

Chapter
Section
Textbook Problem

Evaluating a Sum In Exercises 25–28, use the summation formulas to rewrite the expression without the summation notation. Use the result to find the sums for n = 10,100, 1000, and 10,000. ∑ j = 1 n 7 j + 4 n 2

To determine

To calculate: The sum of the provided expression j=1n7j+4n2 using summation formulas.

Explanation

Given:

The provided expression is: j=1n7j+4n2.

Formula used:

The sum of first n natural is given by the formula:

i=1ni=n(n+1)2

And n times sum of constant is written as:

i=1nc=nc

Calculation:

Start with the provided expression. That is,

j=1n7j+4n2

Split the expression in two parts to use summation formulas,

j=1n7jn2+j=1n4n2

Factor out 7n2 from first summation and 1n2 from second summation,

7n2j=1nj+1n2j=1n4

Now apply the formulas for summation:

i=1ni=n(n+1)2 and i=1nc=nc.

So,

7n2j=1nj+1n2j=1n4=7n2(n(n+1)2)+1n2(n×4)

Simplify,

7n2(n(n+1)2)+1n2<

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