Geometric series

Suppose you and I take turns at rolling adie, to see who can first roll a six. Suppose I roll first, then you roll, andso on, until one of us has rolled a six. What is the chance that you roll thefirst six?
T= number of rolls required to produce the first six.
P(T even) = P(T=2) + P(T=4) + P(T=6) + ...
= qp + q^3p + q^5p +....
= qp (1+ q^2 + q^4 +.....)
= qp / (1-q^2)
Sol..
A=1+q^2+q^4+q^6+….. ..
Aq^2=q^2+q^4+q^6+…. ..
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A(1－q^2)=1
A=1/(1－q^2).

sum to infinity of a geometric series with first term a and common ration r where -1<1 is a/(1-r)