Question

If the numbers $3^{2a-1},14,3^{4-2a}(0<a<1)$ are the first three terms of an $AP$, then its fifth term is equal to

• A. 33
• B. 43
• C. 53
• D. 63

Answer 1

Answer: (C)

By hypothesis $3^{2a-1}+3^{4-2a}=28$. Therefore
$\frac{9^a}{3}+\frac{81}{9^a}=28$
Substituting $9^a=x$, we get $\frac{x}{3}+\frac{81}{x}=28$
$\Rightarrow x^2-84x+243=0$
$\Rightarrow (x-81)(x-3)=0$
$\Rightarrow x=81$ or $x=3$
This gives $9^a=81$ or $9^a=3$
$\Rightarrow a=2$ or $\frac{1}{2}$
$\because 0<a<1$
$\Rightarrow a=\frac{1}{2}$
Therefore, the numbers are $1,14,27$, which are in AP with common difference $13$ The fifth term
$=1+4\times 13=53$