Sequence and series-3

Geometric Mean

· If a, G, b are in G.P., then G is called G.M. between a and b. and G2 = 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 G2 = 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  rk

(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  a1, a2, a3, a4...........an be in G.P.

Then a1 x an = a2 x an-1 = ax an-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 a1, a2, a3, a4...........ais a G.P. of non-zero, non-negative terms,

then log a1, log a2, log a3, log a4...........log ais an A.P. and vice-versa.

(7)  Three non-zero numbers a, b, c are in G.P., iff b2 =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 Sn denotes the sum of the first m terms, then the sum of their product taken two by two is : r SnSn-1/(r+1)

(10)  If  ax,ay,az.....are in G.P., then  x, y, z will be are  in  A.P.