How to find the base of an infinite series?

Question

$$P(k)=\frac c{a^k},k=2,3,4,5,6,\dots$$ Isn't the base of this infinite series $1/a$? I been trying to find the probability of the odd outcome using 1st term/1-base, but I keep getting the wrong answer.

Answer 1

First of all we must have$$\sum_{k=2}^{\infty}\dfrac{c}{a^k}=1$$here we must have $|a|>1$ therefore:$$\sum_{k=2}^{\infty}\dfrac{c}{a^k}=\dfrac{c}{a(a-1)}=1$$After that we have:$$P_{Odd}=c(\dfrac{1}{a^3}+\dfrac{1}{a^5}+\dfrac{1}{a^7}+...)=\dfrac{c}{a^3}(1+\dfrac{1}{a^2}+\dfrac{1}{a^4}+...)=\dfrac{c}{a(a-1)(a+1)}=\dfrac{1}{a+1}$$