# JEE Main & Advanced Mathematics Geometric Progression Relation Between A.P., G.P. and H.P.

## Relation Between A.P., G.P. and H.P.

Category : JEE Main & Advanced

(1) If $A,\,\,G,\,\,\,H$ be A.M., G.M., H.M. between $a$ and $b,$ then

\frac{{{a}^{n+1}}+{{b}^{n+1}}}{{{a}^{n}}+{{b}^{n}}}=\left\{ \begin{align} & A\text{ when }n=0 \\ & G\text{ when }n=-1/2 \\ & H\text{ when }n=-1 \\\end{align} \right.

(2) If ${{A}_{1}},\,{{A}_{2}}$ be two A.M.?s; ${{G}_{1}},\,{{G}_{2}}$ be two G.M.?s and ${{H}_{1}},\,{{H}_{2}}$ be two H.M.?s between two numbers $a$ and $b,$ then

$\frac{{{G}_{1}}{{G}_{2}}}{{{H}_{1}}{{H}_{2}}}=\frac{{{A}_{1}}+{{A}_{2}}}{{{H}_{1}}+{{H}_{2}}}$

(3) Recognization of A.P., G.P., H.P. : If $a,\,\,b,\,\,c$ are three successive terms of a sequence.

If  $\frac{a-b}{b-c}=\frac{a}{a}$, then $a,\,\,b,\,\,c$ are in A.P.

If, $\frac{a-b}{b-c}=\frac{a}{b}$, then $a,\,\,b,\,\,c$ are in G.P.

If, $\frac{a-b}{b-c}=\frac{a}{c}$, then $a,\,\,b,\,\,c$ are in H.P.

(4) If number of terms of any A.P./G.P./H.P. is odd, then A.M./G.M./H.M. of first and last terms is middle term of series.

(5) If number of terms of any A.P./G.P./H.P. is even, then A.M./G.M./H.M. of middle two terms is A.M./G.M./H.M. of first and last terms respectively.

(6) If ${{p}^{th}},\,\,{{q}^{th}}$ and ${{r}^{th}}$ terms of a G.P. are in G.P. Then $p,\,\,q,\,\,r$ are in A.P.

(7) If $a,\,\,b,\,\,c$ are in A.P. as well as in G.P. then $a=b=c$.

(8) If $a,\,\,b,\,\,c$ are in A.P., then ${{x}^{a}},\,{{x}^{b}},\,{{x}^{c}}$ will be in G.P. $(x\ne \pm 1)$.

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