# The value of the telescoping sum ∑ i = 1 n [ i 4 − ( i − 1 ) 4 ] .

### 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 41E

(a)

To determine

## To find: The value of the telescoping sum ∑i=1n[i4−(i−1)4].

Expert Solution

The value of the telescoping sum i=1n[i4(i1)4] is n4.

### Explanation of Solution

Simplify the expression i=1n[i4(i1)4] and obtain the value of the sum.

i=1n[i4(i1)4]=(1404)+(2414)+(3424)++(n4(n1)4)=n404=n4

Thus, the value of the telescoping sum i=1n[i4(i1)4] is n4.

(b)

To determine

### To find: The value of the telescoping sum ∑i=1100[5i−5i−1].

Expert Solution

The value of the telescoping sum i=1100[5i5i1] is 51001.

### Explanation of Solution

Simplify the expression i=1100[5i5i1] and obtain the value of the sum.

i=1100[5i5i1]=(51511)+(52521)+(53531)++(510051001)=(5150)+(5251)+(5352)++(5100599)=510050=51001

Thus, the value of the telescoping sum i=1100[5i5i1] is 51001.

(c)

To determine

### To find: The value of the telescoping sum ∑i=399[1i−1i+1].

Expert Solution

The value of the telescoping sum i=399[1i1i+1] is 97100.

### Explanation of Solution

Simplify the expression i=399[1i1i+1] and obtain the value of the sum.

i=399[1i1i+1]=(1313+1)+(1414+1)+(1515+1)++(199199+1)=(1314)+(1415)+(1516)++(1991100)=131100=97100

Thus, the value of the telescoping sum i=399[1i1i+1] is 97100.

(d)

To determine

### To find: The value of the telescoping sum ∑i=1n(ai−ai−1).

Expert Solution

The value of the telescoping sum i=1n(aiai1) is ana0.

### Explanation of Solution

Simplify the expression i=3n(aiai1) and obtain the value of the sum.

i=1n(aiai1)=(a1a11)+(a2a21)+(a3a31)++(anan1)=(a1a0)+(a2a1)+(a3a2)++(anan1)=ana0

Thus, the value of the telescoping sum i=1n(aiai1) is ana0.

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