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

Question

If $a, b, c,$ are in G.P., prove that the following are also in G.P.
$\text{a}^3,\text{b}^3,\text{c}^3$

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Answer

$a, b, c$ are in G.P.
$a, b = ar, c = ar^2$
$\big(\text{b}^3\big)^2=\text{a}^3\text{c}^3$
$\big((\text{ar})^3\big)^2=\text{a}^3\big(\text{ar}^2\big)^3$
$\text{a}^6\text{r}^6=\text{a}^3\big(\text{a}^3\text{r}^6\big)$
$\text{a}^6\text{r}^6=\text{a}^6\text{r}^6$
$\text{L.H.S}=\text{R.H.S}$
$\Rightarrow\big(\text{b}^3\big)^2=\text{a}^3\text{c}^3$
So,
$\Rightarrow\text{a}^3,\text{b}^3,\text{c}^3\text{ are in G.P.}$

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