Let a and b be two distinct positive real numbers.

Question

Let $a$ and $b$ be two distinct positive real numbers. The 11th term of a G.P. whose first term is $a$ and third term is $b$, is equal to the $p$th term of another G.P. whose first term is $a$ and fifth term is $b$. Then $p$ is equal to:

Accepted Answer

First G.P.: ratio $r$ with $ar^2=b \Rightarrow r^2=b/a$. Its 11th term is $ar^{10} = a(b/a)^5$.

Second G.P.: ratio $r'$ with $ar'^4=b \Rightarrow r'^4=b/a$. Its $p$th term is $ar'^{p-1} = a(b/a)^{(p-1)/4}$.

Equating exponents of $(b/a)$ (since $a
e b$): $5 = \dfrac{p-1}{4} \Rightarrow p = 21.$

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