Geometric progression

Geometric progression series is the one in which T_n+1 term is obtained by multiplying a fix quantity to Tn, known as common ratio.

A sample Geometric progression is a, ar, ar^2, ar^3, ar^4, ar^5…

Important Result:

    For a GM the ratio   Tn+1/Tn = r is always constant. This constant r is known as common ratio.
    The sum of the GM series is

Sum =  a(r^n-1)/((r-1) )

Where
A is the first term of the GM
r is the common ratio
n is the number of terms

    Sum of the GM in terms of the last term and first term
Sum of GM =  (r (last term) – first term)/((r-1))

    Tn = Sn – Sn-1

Where,
    Tn is the nth term of the GM
    Sn is the sum of first nth terms of the GM
    Sn+1 is the sum of first (n-1)th terms of the GM
    If  r, the common ratio of the GM is less than 1 then the sum of the GM for infinite terms is

Sum  = a/(1-r)


Geometric Mean

    The geometric mean of two numbers a and b is √ab
    The geometric mean of n numbers a1, a2, a3, a4, a5, a6, a7, a8, a9,….an is

√(a1 a2 a3 a4 a5 a6 a7 a8 a9….an)

Inserting ‘n’ numbers of geometric means between any two numbers, A and B.

Let the two numbers between which the ‘n’ Geometric means are to inserted to be A and B.
Now there are  (n+2) terms in the series. The first one is A and the last one is B.

Let assume the common ratio is r, then the last term B could be written as
B = A.r⁽ⁿ⁺¹⁾
    r⁽ⁿ⁺¹⁾  = B/A
   r = (B/A)^1/(n+1)

Therefore If we are given the value of A and B and the number of terms ‘n’ then we can easily insert n geometric means between A and B.