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

Vedclass pdf generator app on play store
Vedclass iOS app on app store

It is given that $a, b, c$ and $d$ are in $G.P.$

$\therefore b^{2}=a c$       ........$(1)$

$c^{2}=b d$       ........$(2)$

$a d=b c$       ........$(3)$

It has to be proved that $\left(a^{n}+b^{n}\right),\left(b^{n}+c^{n}\right),\left(c^{n}+d^{n}\right)$ are in $G.P.$ i.e.,

$\left(b^{n}+c^{n}\right)^{2}=\left(a^{n}+b^{n}\right),\left(c^{n}+d^{n}\right)$

Consider $L.H.S.$

$\left(b^{n}+c^{n}\right)^{2}=b^{2 n}+2 b^{n} c^{n}+c^{2 n}$

$=\left(b^{2}\right)^{n}+2 b^{n} c^{n}+\left(c^{2}\right)^{n}$

$=(a c)^{n}+2 b^{n} c^{n}+(b d)^{n}$            [ Using $(1)$ and $(2)$ ]

$=a^{n} c^{n}+b^{n} c^{n}+b^{n} c^{n}+b^{n} d^{n}$

$=a^{n} c^{n}+b^{n} c^{n}+a^{n} d^{n}+b^{n} d^{n}$         [ Using $(3)$ ]

$=c^{n}\left(a^{n}+b^{n}\right)+d^{n}\left(a^{n}+b^{n}\right)$

$=\left(a^{n}+b^{n}\right)\left(c^{n}+d^{n}\right)=$ $\mathrm{R.H.S.}$

$\therefore\left(b^{n}+c^{n}\right)^{2}=\left(a^{n}+b^{n}\right)\left(c^{n}+d^{n}\right)$

Thus, $\left(a^{n}+b^{n}\right),\left(b^{n}+c^{n}\right),$ and $\left(c^{n}+d^{n}\right)$ are in $G.P.$

Similar Questions

If ${a_1},\;{a_2},\;{a_3}.......{a_n}$ are in $A.P.$, where ${a_i} > 0$ for all $i$, then the value of $\frac{1}{{\sqrt {{a_1}} + \sqrt {{a_2}} }} + \frac{1}{{\sqrt {{a_2}} + \sqrt {{a_3}} }} + $ $........ + \frac{1}{{\sqrt {{a_{n - 1}}}  + \sqrt {{a_n}} }} = $

  • [IIT 1982]

If the sum of first $n$ terms of an $A.P.$ be equal to the sum of its first $m$ terms, $(m \ne n)$, then the sum of its first $(m + n)$ terms will be

The sum of $n$ terms of two arithmetic progressions are in the ratio $(3 n+8):(7 n+15) .$ Find the ratio of their $12^{\text {th }}$ terms.

${7^{th}}$ term of an $A.P.$ is $40$, then the sum of first $13$ terms is

If $19^{th}$ terms of non -zero $A.P.$ is zero, then its ($49^{th}$ term) : ($29^{th}$ term) is

  • [JEE MAIN 2019]