Question

If a, b, c are in A.P. b, c, d are in G.P. and $\frac{1}{c},\frac{1}{d},\frac{1}{e}$ are in A.P., prove that a, c, e are in G.p.

Answer 1

a, b, c are in A.P. $\Rightarrow 2b=a+c...........\left(i\right)$

b, c, d are in G.P. $\Rightarrow c^2=bd.........\left(ii\right)$

$\frac{1}{c},\frac{1}{d},\frac{1}{e}$ are in A.P. $\Rightarrow \frac{2}{d}=\frac{1}{c}+\frac{1}{e}.......\left(iii\right)$

We need to prove that

a, b, c are in G.P.

$\Rightarrow c^2=ae$

Now,

$c^2=bd=2b\times \frac{d}{2}$

$\Rightarrow c^2=(a+c)\times \frac{ce}{c+e}$

$[\because \frac{2}{d}=\frac{e+c}{ce}]$

$\Rightarrow c^2=(c+e)=ace+c^{2}e$

$\Rightarrow c^3+c^{2}e=ace+c^{2}e$

$\Rightarrow c^3=ace$

$\Rightarrow c^2=ae$

Hence proved.