In a G.P., t 2 + t 5 = 216 and t 4 : t 6 = 1 : 4 and all terms are integers, then its first term is

Question

In a G.P., $t_{2} + t_{5} = 216$ and $t_{4} : t_{6} = 1 : 4$ and all terms are integers, then its first term is

Smart Achievers · Accepted Answer

(3)

$Let G.P. is a, a r, a r^{2}, a r^{3}, a r^{4}, \dots$

$∴$ $t_{2} + t_{5} = a r + a r^{4} = 216$

$and$ $\frac{t_{4}}{t_{6}} = \frac{a r^{3}}{a r^{5}} = \frac{1}{4} \Rightarrow r^{2} = 4 \Rightarrow r = \pm 2$

$For r = 2$

$For r = - 2$

$\begin{matrix} a (2 + 2^{4}) = 216 \\ \Rightarrow a (18) = 216 \\ \Rightarrow a = \frac{216}{18} = 12 \end{matrix} | \begin{matrix} a (- 2 + 2^{4}) = 216 \\ \Rightarrow a (14) = 216 \\ \Rightarrow a = \frac{216}{14} = \frac{108}{7} \end{matrix}$

$∴$ $a = 12$ $(since all terms are integers)$

Solution

Q.
In a G.P., $t_{2} + t_{5} = 216$ and $t_{4} : t_{6} = 1 : 4$ and all terms are integers, then its first term is

1 16

2 14

3 12

4 none of these

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