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# If a, b, c, d are in GP, what is the progression of $a^n + b^n$, $b^n + c^n$, $c^n + d^n$.

$\begin{array}{1 1} AP \\ HP \\ GP \\ No\;such\;relationship\;exists \end{array}$

$Explanation\;:To\;check\;if\;it\;is\;in\;GP-$
$a=a\;,b=ar\;,c=ar^2\;,d=ar^3$
$if\;a^n+b^n\;,b^n+c^n\;,c^n+d^n\;are\;in\;GP\;,\large\frac{a^n+b^n}{b^n+c^n}=\large\frac{b^n+c^n}{c^n+d^n}$
Lets see if it is true LHS=$\large\frac{a^n+b^n}{b^n+c^n}$
$=\large\frac{a^n+(ar)^n}{(ar)^n+(ar^2)^n}=\large\frac{a^n(1+r^n)}{a^n(r^n+r^{2n})}$
$=\large\frac{1+r^n}{r^n+r^{2n}}$
$RHS=\large\frac{b^n+c^n}{c^n+d^n}=\large\frac{(ar)^n+(ar^2)^n}{(ar^3)^n+(ar^2)^n}$
$=\large\frac{a^n(r^n+r^{2n})}{a^n(r^{3n}+r^{2n})}=\large\frac{r^n+r^{2n}}{r^{3n}+r^{2n}}$
$if\;LHS=RHS$
$\large\frac{1+r^n}{r^n+r^{2n}}=\large\frac{r^n+r^{2n}}{r^{3n}+r^{2n}}$
$(1+r^n)(r^{3n}+r^{2n})=(r^n+r^{2n})(r^n+r^{2n})$
$LHS=r^{3n}+r^{3n+n}+r^{2n}+r^{2n+n}$
$=r^{3n}+r^{4n}+r^{2n}+r^{3n}$
$RHS=r^{2n}+r^{2n+n}+r^{2n+n}+r^{2n+2n}$
$=r^{2n}+r^{3n}+r^{3n}+r^{4n}$
$Since\;LHS\;=\;RHS\;,the\;expressions\;are\;in\;GP.$
edited Jan 24, 2014