If $a, b, c, d$ are in G.P., show that
i) $a^{2} + b^{2}, b^{2} + c^{2}, c^{2} + d^{2}$ are in G.P

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Solution

If $a, b, c, d$ are in G.P. , then
$a = a$
$b = a r$
$c = a r^{2}$
$d = a r^{3}$

where $a$ is the first term and $r$ is the common ratio.

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

$b^{2} + c^{2} = a^{2} r^{2} + a^{2} r^{4} = a^{2} (1 + r^{2}) \times r^{2}$

$c^{2} + d^{2} = a^{2} r^{4} + a^{2} r^{6} = a^{2} (1 + r^{2}) \times r^{4}$

which are in G.P. with
first term, $A = a^{2} (1 + r^{2})$
and common ratio, $R = r^{2}$

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