If a, b, c, d are in G.P, prove that $(a^{n} + b^{n}), (b^{n} + c^{n}), (c^{n} + d^{n})$ are in G.P.

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Solution

We know that $a, a r, a r^{2}, a r^{3}, . . . . . .$ are in $G . P .$
$a = a$
$b = a r$
$c = a r^{2}$
$d = a r^{3}$
To show $(a^{n} + b^{n}), (b^{n} + c^{n}), (c^{n} + d^{n})$
i.e $\frac{b^{n} + c^{n}}{a^{n} + b^{n}} = \frac{c^{n} + d^{n}}{b^{n} + c^{n}}$
Take LHS Take RHS
$\frac{b^{n} + c^{n}}{a^{n} + b^{n}}$ put $log b = a r, c = a r^{2}$ $= \frac{c^{n} + d^{n}}{b^{n} + c^{n}}$
$= \frac{(a r)^{n} + (a r^{2})^{n}}{a^{n} + (a r)^{2}}$ $= \frac{(a r^{2})^{n} + (a r^{3})^{n}}{(a r)^{n} + ((a r)^{2})^{n}}$
$= \frac{a^{n} r^{n} + a^{n} r^{2 n}}{a^{n} + a^{n} r^{n}}$ $= \frac{a^{n} (r^{2 n} + r^{3 n})}{a^{n} (r^{n} + r^{2 n})}$
$= \frac{a^{n} r^{n} (1 + r^{n})}{a^{n} (1 + r^{n})}$ $= \frac{r^{2 n} (1 + r)}{r^{n} (1 + r)}$
$= r^{n}$ $= r^{n}$
$∴$ LHS=RHS
$∴ (a^{n} + b^{n}), (b^{n} + c^{n}), (c^{n} + d^{n})$ are in $G P$ .

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