# JEE Main & Advanced Mathematics Geometric Progression Properties of G.P.

## Properties of G.P.

Category : JEE Main & Advanced

(1) If all the terms of a G.P. be multiplied or divided by the same non-zero constant, then it remains a G.P., with the same common ratio.

(2) The reciprocal of the terms of a given G.P. form a G.P. with common ratio as reciprocal of the common ratio of the original G.P.

(3) If each term of a G.P. with common ratio r be raised to the same power k, the resulting sequence also forms a G.P. with common ratio ${{r}^{k}}$.

(4) In a finite G.P., the product of terms equidistant from the beginning and the end is always the same and is equal to the product of the first and last term. i.e., if ${{a}_{1}},\,{{a}_{2}},\,{{a}_{3}},\,......\,{{a}_{n}}$ be in G.P.

Then ${{a}_{1}}\,{{a}_{n}}={{a}_{2}}\,{{a}_{n-1}}={{a}_{3}}\,{{a}_{n-2}}={{a}_{4}}\,{{a}_{n-3}}=..........={{a}_{r}}\,.\,{{a}_{n-r+1}}$

(5) If the terms of a given G.P. are chosen at regular intervals, then the new sequence so formed also forms a G.P.

(6) If ${{a}_{1}},\,{{a}_{2}},\,{{a}_{3}},\,.....,\,{{a}_{n}}......$ is a G.P. of non-zero, non-negative terms, then $\log {{a}_{1}},\,\log {{a}_{2}},\,\log {{a}_{3}},\,.....\log {{a}_{n}},\,......$ is an A.P. and vice-versa.

(7) Three non-zero numbers a, b, c are in G.P., iff ${{b}^{2}}=ac$.

(8) If first term of a G.P. of $n$ terms is $a$ and last term is $l,$ then the product of all terms of the G.P. is ${{(al)}^{n/2}}$.

(9) If there be $n$ quantities in G.P. whose common ratio is $r$ and ${{S}_{m}}$ denotes the sum of the first m terms, then the sum of their product taken two by two is $\frac{r}{r+1}\,{{S}_{n}}\,{{S}_{n-1}}$.

(10) If ${{a}^{{{x}_{1}}}},{{a}^{{{x}_{2}}}},{{a}^{{{x}_{3}}}},....,{{a}^{{{x}_{n}}}}$ are in G.P., then ${{x}_{1}},{{x}_{2}},{{x}_{3}},....,{{x}_{n}}$ will be are  in  A.P. ,

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