Question

The problem is:Prove that the statement 1/5+1/5^2+1/5^3+...1/5^n=1/4(1-1/5^n) is true for all positive integers n. Write your proof in the space below.

How do I start this? I have looked at the only example in the book but it did not help me.

Any help in this would be great!!

Answer 1

Here is the answer:

Let S represent the sum of the expression
then:

S = 1/5 + 1/5^2 +... + 1/5^n ---- (a)

multiply both sides by 1/5

(1/5)S = 1/5^2 + 1/5^3 +.. + 1/5^n + 1/5^(n+1) ---- (b)

use equation (a) subtract equation (b)

(4/5)S = 1/5 - 1/5^(n+1)

multiply 5 on both sides:

4S = 1 - 1/5^n

divide 4 on both sides:

S = (1/4)(1-1/5^n)

that's the proof.