Question

The numbers $1,4,16$ can be three terms (not necessarily consecutive) of

• A. no AP
• B. only one GP
• C. infinite number of APs
• D. infinite number of GPs

Answer 1

Answer: (C), (D)

infinite number of GPs

$4=1+(n-1)d,16=1+(m-1)d\Rightarrow \frac{15}{3}=\frac{m-1}{n-1}$
$\Rightarrow \frac{n-1}{1}=\frac{m-1}{5}=p=$ positive integer

$\therefore n=p+1,m=5p+1$.
So, $n,m$ have infinite pairs of values.

Also, $4=1\cdot r^n,16=1\cdot r^m$
$r^{2n}=r^m$
$\Rightarrow m=2n$
$\therefore \frac{m}{2}=\frac{n}{1}=q=$ positive integer
So, $m,n$ have infinite pairs of values.