Question

In a GP, $t_2+t_5=216$ and $t_4:t_6=1:4$ and all terms are integers, then its first term is

• A. $16$
• B. $14$
• C. $12$
• D. None of the above

Answer 1

The correct option is C

$12$

Explanation for the correct answer:

To find the first term of GP.

In a GP, the formula for $n^{th}$ term is $t_n=a{r}^{n-1}$ where, $a$ is the first term and $r$ is the common ratio

So,

$t_2=ar,t_5=a{r}^4,t_4=a{r}^3,t_6=a{r}^5$

Given,

$\begin{array}{rcl}{t}_2+t_5& =& 216\\ ar+a{r}^4& =& 216\\ ar\left(1+r^3\right)& =& 216\cdots \cdots \left(1\right)\end{array}$

And,

$\begin{array}{rcl}\frac{t_4}{t_6}& =& \frac{1}{4}\\ \frac{a{r}^3}{a{r}^5}& =& \frac{1}{4}\\ r^2& =& 4\\ r& =& \pm 2\end{array}$

at,

$\begin{array}{rcl}r& =& 2\\ 2a\left(1+8\right)& =& 216\mathbf{[}\mathbf{\because }\mathbf{from}\mathbf{}\mathbf{1}\mathbf{]}\\ a& =& \frac{216}{18}=12\end{array}$

at,

$\begin{array}{rcl}r& =& -2\\ -2a\left(1-8\right)& =& 216\\ a& =& \frac{216}{14}\\ & & \end{array}$

Given all terms are integers, So $r$ must be $2$and $a$ must be $12$

Hence, the correct option is C.