Let a_1,a_2,a_3, be a G.P.

Question

Let $a_1,a_2,a_3,\ldots$ be a G.P. of increasing positive numbers. If the product of the fourth and sixth terms is $9$, and the sum of the fifth and seventh terms is $24$, then $a_1a_9+a_2a_4a_9+a_5+a_7$ is equal to \_\_\_\_\_\_\_\_.

Answer: 60

Accepted Answer

$a_4a_6 = a_5^2 = 9 \Rightarrow a_5 = 3$ (positive terms). $a_5+a_7 = a_5(1+r^2) = 24 \Rightarrow 1+r^2=8 \Rightarrow r^2=7$.

With $a_5=ar^4=3$, $a = \dfrac{3}{49}$.

$a_1a_9 = a^2r^8 = \dfrac{9}{2401}\times2401 = 9$; $a_2a_4a_9 = a^3r^{12} = \dfrac{27}{117649}\times117649 = 27$.

$$a_1a_9+a_2a_4a_9+a_5+a_7 = 9+27+3+21 = 60.$$

Solutions · 1