Question

If $a,b,c$ are in A.P., then $3^{\mathrm{a}},3^{\mathrm{b}},3^{\mathrm{c}}$ shall be in

Answer 1

Finding the progression of the given expression:

Given that $a,b,c$ are in AP

$\Rightarrow$$b-a=c-b$

Apply $3^x$ on both sides
$\Rightarrow$ $3^{b-a}=3^{c-b}$

$\Rightarrow$ $\frac{3^b}{3^a}=\frac{3^c}{3^b}$

$\Rightarrow$${(3^b)}^2=3^a\times 3^c$

$\therefore$$3^{\mathrm{a}},3^{\mathrm{b}},3^{\mathrm{c}}$ are in GP.

Hence, the terms $\mathbf{}{\mathbf{3}}^{\mathbf{a}}\mathbf{,}\mathbf{}{\mathbf{3}}^{\mathbf{b}}\mathbf{,}\mathbf{}{\mathbf{3}}^{\mathbf{c}}$ are in Geometric Progression (GP)