Sequences & Series
GP — product and sum conditions
nta_pyq_2023_jan
Grade 11
Question:
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 ______.
Step-by-Step Solution
Key Concept: From $a_4 \cdot a_6 = a_5^2 = 9 \Rightarrow a_5 = 3$. From $a_5 + a_7 = 24$: $3 + 3r^2 = 24 \Rightarrow r^2 = 7 \Rightarrow r = \sqrt{7}$. Then find first term $a$ and compute.
$a_5 = 3$, $r = \sqrt{7}$. $a_1 a_9 = a_5^2 = 9$. $a_2 a_4 a_9$: using $a_2 = a_5/r^3$, $a_4 = a_5/r$, $a_9 = a_5 r^4$, product $= a_5^3 = 27$. $a_5 + a_7 = 3 + 21 = 24$. Sum $= 9 + 27 + 3 + 21 = 60$.
Correct Answer: 60