Question

If a, b, c, d are in G.P., prove that $\frac{1}{a^2+b^2},\frac{1}{b^2+c^2},\frac{1}{c^2+d^2}$ are in G.P.

Answer 1

Let $b=ar,c=a{r}^2,d=a{r}^3$

$\Rightarrow \frac{1}{c^2+d^2}÷\frac{1}{b^2+c^2}$

$\Rightarrow \frac{b^2+c^2}{c^2+d^2}=\frac{a^2{r}^2+a^2{r}^4}{a^2{r}^4+a^2{r}^6}$

$\Rightarrow \frac{1+r^2}{r^2+r^4}$

$\Rightarrow \frac{1}{b^2+c^2}÷\frac{1}{a^2+b^2}$

$\Rightarrow \frac{a^2+a^2{r}^2}{a^2{r}^2+a^2{r}^4}$

$\Rightarrow \frac{1+r^2}{r^2+r^4}$

Common ratio is same hence proved.