**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. ,

*play_arrow*Definition*play_arrow*General term of a G.P.*play_arrow*Selection of Terms in a G.P.*play_arrow*Sum of first 'n' terms of a G.P.*play_arrow*Sum of infinite terms of a G.P.*play_arrow*Geometric Mean*play_arrow*Properties of G.P.*play_arrow*Definition*play_arrow*\[{{n}^{th}}\]term of A.G.P.*play_arrow*Sum of A.G.P.*play_arrow*Method for Finding Sum*play_arrow*Method of Difference*play_arrow*Special Series*play_arrow*Properties of Arithmetic, Geometric, Harmonic Means Between Two Given Numbers*play_arrow*Relation Between A.P., G.P. and H.P.

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