Question

Let \( a_1, a_2, a_3, \dots \) be a G.P. of increasing positive terms. If \( a_1 a_5 = 28 \) and \( a_2 + a_4 = 29 \), then the value of \( a_6 \) is equal to:

• A. 628
• B. 526
• C. 784
• D. 812

Hint:

In geometric progressions, use the common ratio \(r\) and apply the equations involving terms to find the missing terms.

Answer 1

The Correct Option is C

$a_1 a_5 = 28 \Rightarrow a \cdot ar^4 = 28 \Rightarrow a^2 r^4 = 28 \quad \text{...(1)}$ 
$a_2 + a_4 = 29 \Rightarrow ar + ar^3 = 29$ $\Rightarrow ar(1+r^2) = 29$ $\Rightarrow a^2 r^2 (1+r^2)^2 = (29)^2 \quad \text{...(2)}$ By Eq. (1) \& (2) $\frac{r^2}{(1+r^2)^2} = \frac{28}{29 \times 29}$ 
$\frac{r}{1+r^2} = \frac{\sqrt{28}}{29} \Rightarrow r = \sqrt{28}$ 
$\therefore a^2 r^4 = 28 \Rightarrow a^2 \times (28)^2 = 28$ 
$a = \frac{1}{\sqrt{28}}$ 
$\therefore a_6 = ar^5 = \frac{1}{\sqrt{28}} \times (28)^2 \sqrt{28} = 784$