If a, b, c are in G.P., prove taht the following are also
i) $a^{3}, b^{3}, c^{3}$
ii) $a^{2} + b^{2}, a b + b c +, b^{2} + c^{2}$

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Solution

$a, b, c$ are in G.P.
Let the common ratio is $r$

Then
$b = a r$

and
$c = a r^{2}$

(1)

First term, $A_{1} = a^{3}$

Second term, $A_{2} = b^{3} = (a r)^{3} = a^{3} r^{3}$

Third term, $A_{3} = c^{3} = (a r^{2})^{3} = a^{3} r^{6}$

Here, $\frac{A_{2}}{A_{1}} = \frac{A_{3}}{A_{2}} = r^{3}$

so, these are in G.P. with common ratio $r^{3}$

(2)

$A_{1} = a^{2} + b^{2} = a^{2} + a^{2} r^{2}$
$= a^{2} (1 + r^{2})$

$A_{2} = a b + b c = b (a + c)$
$= a r (a + a r^{2})$
$= a^{2} r (1 + r^{2})$

$A_{3} = b^{2} + c^{2} = a^{2} r^{2} + a^{2} r^{4}$
$= a^{2} r^{2} (1 + r^{2})$

Here, $\frac{A_{2}}{A_{1}} = \frac{A_{3}}{A_{2}} = r$

so, these are in G.P. with common ratio $r$

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Similar questions

If a, b, c are in G.P., prove that:

(i) $a (b^{2} + c^{2}) = c (a^{2} + b^{2})$

(ii) $a^{2} b^{2} c^{2} (\frac{1}{a^{3}} + \frac{1}{b^{3}} + \frac{1}{c^{3}}) = a^{3} + b^{3} + c^{3}$

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