Sequences & Series
AP and GP conditions leading to abc
nta_pyq_2023_jan
Grade 11
Question:
Let $a, b, c > 1$, $a^3$, $b^3$ and $c^3$ be in A.P., and $\log_a b$, $\log_c a$ and $\log_b c$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is $\dfrac{a+4b+c}{3}$ and the common difference is $\dfrac{a-8b+c}{10}$ is $-444$, then $abc$ is equal to
343
216
$\dfrac{343}{8}$
$\dfrac{125}{8}$
Step-by-Step Solution
Key Concept: From the GP of logarithms: $a = c$. Substituting into the AP condition $a^3 + c^3 = 2b^3$ with $a=c$ gives $a = b = c$. Then use $S_{20} = -444$ to find $a$.
$a = b = c = 6$, so $abc = 216$.
Correct Answer: 2