QJEEM23Ma2adf: JEE Main 2023 Mathematics - Class 11 JEEM23Ma2adf Solution: Let and be in A.P., and and be in G.P. If the sum of first 20…

Question

Verified Solution • 2025–26 Session moderate

Let

a, b, c&gt;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

\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 :

Accepted Answer

$2{b^3} = {a^3} + {c^3}$

${\left( {{{\log a} \over {\log c}}} \right)^2} = \left( {{{\log b} \over {\log a}}} \right)\left( {{{\log c} \over {\log b}}} \right)$

$\Rightarrow {(\log a)^3} = {(\log c)^3}$

$\Rightarrow \log a = \log c$

$\Rightarrow a = c$

$\Rightarrow a = b = c$

${T_1} = 2a,d = - {{3a} \over 5}$

${S_{20}} = - 444$

$\Rightarrow {{20} \over 2}\left( {2(2a) + (19)\left( { - {{3a} \over 5}} \right)} \right) = - 444$

$\Rightarrow 10{{(20a - 57a)} \over 5} = - 444$

$\Rightarrow 37a = 222$

$\Rightarrow a = 6$

$\Rightarrow abc = {(6)^3} = 216$

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