Geometric Progression

In mathematics, a geometric progression, are called as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54 ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. The sum of the conditions of a geometric progression

Source: Wikipedia

Concepts of geometric progression:

A geometric progression is a sequence of number each of which, after the first, is obtained by multiplying the preceding number by a constant r. This constant r is called the common ration of the geometric progression.

In other words, a series is said in geometric progression if the ratio of any term to the preceding term is constant throughout. It is briefly written as geometric progression.

The constant ratio is called the common ratio. It is denoted by “r”.

Thus, the general form of a geometric sequence is

a,ar,ar2,ar3,ar4,…

And that of a geometric series is

a+ar+ar2+ar3+ar4+…

Where r ≠ 0 is the common ratio and a is a scale factor, equal to the sequence's start value.

Formula for geometric progression:

For the geometric progression represented by

a,ar,ar2,ar3,…,arn-1

With common ratio r, the sum of the first n terms, denoted by Sn is

Sn=a+ar+ar2+ar3+…+arn-1

Multiplying both sides by r, we get

rSn=ar+ar2+ar3+…+arn-1+arn

Subtracting, we have

Sn-rSn=a-arn

Sn(1-r)=a(1-rn)

So that Sn= a(1-rn)/(1-r)       if r≠1.

When r=1, the progression is merely a,a,…a so that Sa in this case is simply na. Note that Sn can also be written as

Sn= (r_(n-1))/(r-1)                    r≠1

Example 1:

Find the 12th term of the series:

3,6,12,24,48…

Solution:

The given series is a geometric progression with a=3,r=6/2 =2

T12=ar11=3 x 211=6144

Practice problem for geometric progression:

The sum of the first 8 terms of the geometric progression with the first term a=25 and the common ration r=- 1/5 is

Answer: S8=25 (1-(1/(5_8) )) / (6/5 )