Let a, b, c1, a^3, b^3 and c^3 be in A.P., and _a b, _c a and _b c be in G.P.

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 $\frac{a+4 b+c}{3}$ and the common difference is $\frac{a-8 b+c}{10}$

Accepted Answer

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

${(\frac{\log a}{\log c})}^{2}=(\frac{\log b}{\log a})(\frac{\log c}{\log b})$

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

$\Rightarrow \log a=\log c$

$\Rightarrow a=c$

$\Rightarrow a=b=c$

${T}_{1}=2a,d=-\frac{3a}{5}$

${S}_{20}=-444$

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

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

$\Rightarrow 37a=222$

$\Rightarrow a=6$

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

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