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!!
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.