If a, b, c, are in G.P., prove that the following are also in G.P. a^2+b^2,ab+bc,b^2+c^2

a, b, c are in G.P. a, b = ar, c = ar^2 (ab+bc)^2= (a^2+b^2 ) (b^2+c^2 ) (a +ar ^2 )^2 (a^2+(ar)^2 ) ((ar)^2+ (ar^2 )^2 ) (a^2r+a^2r^3 )^2= (a^2+a^2r^2 ) (a^2r^2+a^2r^4 ) a^4 (r+r^3 )^2=a^4 (1+r^2 ) (r^2+r^4 ) a^4 (r+r^2 )^2=a^4 (1+r^2 )r^2 (1+r^2 ) a^4r^2 (1+r^2 )^2=a^4 (1+r^2 )r^2 (1+r^2 ) a^4r^2 (1+r^2 )^2=a^4r^2 (1+r^2 ) L.H.S=R.H.S (ab+bc)^2= (a^2+b^2 ) (b^2+c^2 ) (a^2+b^2 ),(ab+bc), (b^2+c^2 ) are in G.P.

If a, b, c, are in G.P., prove that the following are also in G.P…

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Question

If $a, b, c,$ are in G.P., prove that the following are also in G.P.
$\text{a}^2+\text{b}^2,\text{ab}+\text{bc},\text{b}^2+\text{c}^2$

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Answer

$a, b, c$ are in G.P.
$a, b = ar, c = ar^2$
$(\text{ab}+\text{bc})^2=\big(\text{a}^2+\text{b}^2\big)\big(\text{b}^2+\text{c}^2\big)$
$\Big(\text{a}\times\text{ar}+\text{ar}\times\text{ar}^2\big)^2\big(\text{a}^2+(\text{ar})^2\big)\big((\text{ar})^2+\big(\text{ar}^2\big)^2\Big)$
$\big(\text{a}^2\text{r}+\text{a}^2\text{r}^3\big)^2=\big(\text{a}^2+\text{a}^2\text{r}^2\big)\big(\text{a}^2\text{r}^2+\text{a}^2\text{r}^4\big)$
$\text{a}^4\big(\text{r}+\text{r}^3\big)^2=\text{a}^4\big(1+\text{r}^2\big)\big(\text{r}^2+\text{r}^4\big)$
$\text{a}^4\big(\text{r}+\text{r}^2\big)^2=\text{a}^4\big(1+\text{r}^2\big)\text{r}^2\big(1+\text{r}^2\big)$
$\text{a}^4\text{r}^2\big(1+\text{r}^2\big)^2=\text{a}^4\big(1+\text{r}^2\big)\text{r}^2\big(1+\text{r}^2\big)$
$\text{a}^4\text{r}^2\big(1+\text{r}^2\big)^2=\text{a}^4\text{r}^2\big(1+\text{r}^2\big)$
$\text{L.H.S}=\text{R.H.S}$
$\Rightarrow(\text{ab}+\text{bc})^2=\big(\text{a}^2+\text{b}^2\big)\big(\text{b}^2+\text{c}^2\big)$
$\Rightarrow\big(\text{a}^2+\text{b}^2\big),(\text{ab}+\text{bc}),\big(\text{b}^2+\text{c}^2\big)\text{ are in G.P.}$