# The number n such that ∑ i = 1 n i = 78 . ### 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 36E
To determine

## To find: The number n such that ∑i=1ni=78.

Expert Solution

The number n is 12.

### Explanation of Solution

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 theorem, it is given that i=1ni=n(n+1)2.

Thus, the expression i=1ni=78 simplified as, n(n+1)2=78.

Solve the equation n(n+1)2=78 and obtain the value of n.

n(n+1)2=78n(n+1)=156n2+n156=0

The equation n2+n156=0 is a quadratic equation with variable n.

Here, the value of a=1,b=1andc=156.

n=b±b24ac2a=1±124(1)(156)2(1)=1±6252=1±252=12or13

That is, the number n is 12or13.

The sum i=1ni=78 is positive thus, the number n=13 is not possible.

Therefore, the sum i=1ni=78 as when n=12.

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