Question

Let $a, b, c>, a^3, b^3$ and $c^3$ be in AP, and $\log _a b, \log _c a$ and $\log _b c$ be in GP If the sum of first 20 terms of an AP, whose first term is $\frac{a+4 b+c}{3}$ and the common difference is $\frac{a-8 b+c}{10}$ is $-444$, then $a b c$ is equal to:

• A. 343
• B. 216
• C. $\frac{343}{8}$
• D. $\frac{125}{8}$

Answer 1

Answer: (B)

216

Answer 2

As a3,b3,c3 be in A.P. →a3+c3=2b3.... (1)
logab​,logca​,logbc​ are in G.P.
∴logalogb​⋅logblogc​=(logcloga​)2
∴(loga)3=(logc)3⇒a=c......(2)
From (1) and (2)
a=b=c
T1​=3a+4b+c​=2a;d=10a−8b+c​=10−6a​=5−3​a
∴S20​=220​[4a+19(−53​a)]
=10[520a−57a​]
=−74a
∴−74a=−444⇒a=6
∴abc=63=216
So, the correct option is (B) : 216