Question

Let $a,b,c$ are $7^{th},{11}^{th}$ and ${13}^{th}$ terms of non constant AP. If $a,b,c$ are also in GP, then find $\frac{a}{c}$

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is D $4$

Let, $p,d$ be the first term and the common difference of the A.P.

Then according to the problem,

$a=p+6d$, $b=p+10d$ and $c=p+12d$.

Given also $a,b,c$ are in G.P. then,

$b^2=ac$

or, $(p+10d{)}^2=(p+6d\left)\right(p+12d)$

or, $p^2+20pd+100d^2=p^2+18pd+72d^2$

or, $28d^2+2pd=0$

or, $p=-14d$. [ Since $d\ne 0$ for the given A.P. to be non-constant]

Now,

$\frac{a}{c}=\frac{-8d}{-2d}=4.$ [ Using the value of $p$]