If the arithmetic mean and geometric mean of the pth and qth terms of the sequence -16, 8,…

Question

If the arithmetic mean and geometric mean of the $p$th and $q$th terms of the sequence $-16, 8, -4, 2, \ldots$ satisfy the equation $4x^2-9x+5=0$, then $p+q$ is equal to \_\_\_\_\_\_\_\_.

Answer: 10

Accepted Answer

$4x^2-9x+5=0 \Rightarrow (x-1)(4x-5)=0 \Rightarrow x=1,\dfrac54$. Since AM$\ge$GM, AM$=\dfrac54$, GM$=1$.

The G.P. has ratio $-	frac12$; the even-indexed terms are positive: $t_{2m}=32/4^m$, giving $t_2=8, t_4=2, t_6=0.5,\ldots$

Checking $t_4+t_6=2.5=$AM$	imes2$ and $t_4t_6=1=$GM$^2$: matches with $p=4,q=6$.

$p+q = 10.$

Solutions · 1