Sequences And Series Question 6
Question 6 - 29 January - Shift 1
Let $a_1, a_2, a_3, \ldots .$. be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24 , then $a_1 a_9+a_2 a_4 a_9+a_5+a_7$ is equal to _________
Show Answer
Answer: 60
Solution:
Formula: General term of a G.P. (I)
$a_4 \cdot a_6=9 \Rightarrow(a_5)^{2}=9 \Rightarrow a_5=3$
& ; $ a_5+a_7=24 \Rightarrow a_5+a_5 r^{2}=24 \Rightarrow(1+r^{2})=8 \Rightarrow r=\sqrt{7}$
$\Rightarrow a=\frac{3}{49}$
$\Rightarrow a_1 a_9+a_2 a_4 a_9+a_5+a_7=9+27+3+21=60$
