· If *a*, *G*, *b* are in G.P., then *G* is called G.M. between *a* and *b*. and G^{2} = ab.

If a,G1,G2, G3,.................Gn,b then G1,G2, G3,.................Gn are n geometric means between a and b.

** ****Insertion of geometric means :**

**Single G.M. between a and b:**

If *a* and *b* are two real numbers then single G.M. between *a* and *b is given by G where *G^{2} = ab

*n* G.M.’s between *a* and *b*:

a,G1,G2, G3,.................Gn,b then G1,G2, G3,.................Gn are n geometric means between a and b

then

**Properties of G.P.**

**(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_{4}...........a_{n} be in G.P.

Then a_{1} x a_{n} = a_{2} x a_{n-1} = a_{3 }x a_{n-2} =...........

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

**(6) **If a_{1}, a_{2}, a_{3}, a_{4}...........a_{n }is a G.P. of non-zero, non-negative terms,

then log a_{1}, log a_{2}, log a_{3}, log a_{4}...........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_{n} denotes the sum of the first *m* terms, then the sum of their product taken two by two is :** r S _{n}S_{n-1}/(r+1)**

**(10) **If a^{x},a^{y},a^{z}.....are in G.P., then x, y, z will be are in A.P.