If a, b, c are in G.P., prove that: (a+2b+2c)(a-2b+2c)=a^2+4c^2 - Vidyadip

Question

If $a, b, c$ are in G.P., prove that:
$(\text{a}+2\text{b}+2\text{c})(\text{a}-2\text{b}+2\text{c})=\text{a}^2+4\text{c}^2$

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Answer

$a, b, c$ are in G.P.$a, b = ar, c = ar^2$
$\text{L.H.S}=({\text{a}+2\text{b}+2\text{c})}{(\text{a}-2\text{ar}+2\text{c})}$
$=\big(\text{a}+2\text{ar}+2\text{ar}^2\big)\big(1-2\text{ar}+2\text{ar}^2\big)$
$=\text{a}^2\big(1+2\text{ar}+2\text{ar}^2\big)\big(1-2\text{r}+2\text{r}^2\big)$
$=\text{a}^2\Big[\big(1+2\text{r}^2\big)^2-\big(2\text{r}\big)^2\Big]$
$=\text{a}^2\big[1+4\text{r}^4+4\text{r}^2-4\text{r}^2\big]$
$=\text{a}^2\big[1+4\text{r}^4\big]$
$=\text{a}^2+4\big(\text{ar}^2\big)^2$
$=\text{a}^2+4\text{c}^2$
$=\text{R.H.S}$
$\therefore\text{R.H.S}=\text{L.H.S}$

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