Question

If a, b, c, d are in G.P., prove that $(a^2+b^2+c^2),(ab+bc+cd),(b^2+c^2+d^2)$ are in G.P.

Answer 1

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

$\Rightarrow \frac{b^2+c^2+d^2}{ab+bc+cd}$

$\Rightarrow \frac{a^2{r}^2+a^2{r}^4+a^2{r}^6}{a^{2}r+a^2{r}^3+a^2{r}^5}$

$\Rightarrow \frac{r+r^3+r^5}{1+r^2+r^4}$

$\Rightarrow \frac{ab+bc+cd}{a^2+b^2+c^2}$

$\Rightarrow \frac{a^{2}r+a^2{r}^3+a^2{r}^5}{a^2+a^2{r}^2+a^2{r}^4}$

$\Rightarrow \frac{r+r^3+r^5}{1+r^2+r^4}$

Some common difference .

Hence, proved.