Question

If the arithmetic mean and geometric mean of the \(p^{th}\) and \(q^{th}\) terms of the sequence \(-16,8,-4,2,\dots\) satisfy the equation \(4x^2-9x+5=0\), then \(p+q\) is equal to __________.

Hint:

The roots of \(4x^2-9x+5=0\) are \(1\) and \(\frac54\); GM \(\le\) AM, so GM \(=1\), AM \(=\frac54\). Both terms must be positive, i.e. even-numbered.

Answer 1

Correct Answer: 10

The sequence is a G.P. with first term \(-16\) and ratio \(-\frac12\): \(a_k=-16\left(-\frac12\right)^{k-1}\).
Roots of \(4x^2-9x+5=0\): \(x=1,\frac54\). Since AM \(\ge\) GM for positive numbers, AM \(=\frac54\) and GM \(=1\). AM \(>0\) and a real GM require both terms positive, so \(p,q\) are even, where \(a_k=16\left(\frac12\right)^{k-1}=2^{5-k}\).
GM: \(\sqrt{a_pa_q}=1\Rightarrow2^{10-p-q}=1\Rightarrow p+q=10\).
AM check: \(2^{5-p}+2^{p-5}=\frac52\Rightarrow2^{5-p}=2\) or \(\frac12\Rightarrow p=4,q=6\) (both even, valid).
Answer: 10