GEOMETRIC PROGRESSION

GEOMETRIC PROGRESSION (GP)

Geometric Progression(GP) or Geometric Sequence is sequence of non-zero numbers in which the ratio of any term and its preceding term is always constant.

For example:

3,6,12,24......is an GP because the common ratio (6/3=12/6=24/12=2) is constant.

General representation of GP:

a,ar²,ar³.....

here a is the first term and r is common ratio of the GP

• nth term of a GP = a rn-1
• Geometric Mean = nth root of product of n terms in the GP
• Sum of ‘n’ terms of a GP (r < 1) = [a (1 – rn)] / [1 – r]
• Sum of ‘n’ terms of a GP (r > 1) = [a (rn – 1)] / [r – 1]
• Sum of infinite terms of a GP (r < 1) = (a) / (1 – r)

Given x=1+a+a2+a3+⋯ and y=1+b+b2+b3+⋯, where a and b are proper fractions, what does 1+ab+a2b2+a3b3+⋯ equal to?

Sum of first infinite series

x=1/(1-a)..............1.

where 1 is the first term and a is the common ratio.

Sum of second infinite series

y=1/(1-b)..............2.

where 1 is the first term and b is the common ratio.

Now the sum of the third infinite series(s)

s=1/(1-ab)

where 1 is the first term and ab is the common ratio.

from 1.

1-a=(1/x)

a=(x-1)/x

and 

b=(y-1)/y

s=1/(1-ab)

s=1/(1-((x-1)(y-1))/xy

or

s=xy/(xy-(xy-x-y+1)

s=xy/(xy-xy+x+y-1)

s=xy/(x+y-1)