# Geometric Progression

A succession of numbers is said to be in a geometric progression if the ratio of any term and the previous term is constant. This constant ratio which is common to any of the two terms is known as the common ratio and is denoted by ‘r’.

Example
1. 1, 2, 4, 8,.,.,.,.,.,.,.,.,.,
2. 20, 10, 5,.,.,.,.,.,
3. aarar2, ……
Common ratio of series (i) is 2.
Common ratio of series (ii) is 0.5.
Common ratio of series (iii) is r.

nth term of a geometric progression

First term t1 = a = ar1-1
Second term t2 = ar = ar2-1
Third term t3 = ar2 = ar3-1
Fourth term t4 = ar3 = ar4-1
nth term tn = arn-1

where a is the first term, r is the common ratio and n is the number of terms.

Important points
• tn is also known as the general term of GP.
• In any question, if some particular term is given like t4 or t10, then we should assume those terms in the form of tn. However, if the total number of terms are given then we should assume the terms in the following way:
If three terms or any odd number of terms are involved, then we should assume these terms asaar and so on.

Example-8
The seventh term of a GP is 8 times the fourth term and the 5th term of the same GP is 48. Find the 6th term of this GP.
Solution
Given t7 = 8 × t4,
Or, ar6 = 8 × ar3
⇒ r3 = 8, or, r = 2
Now, ar4 = 48, So, a = 3
So, 6th term = ar5 = 3 × 25 = 96

# Properties of GP

If a, b, c, d,…are in GP, then
1. ak, bk, ck, dk…will be in GP, where k is any non-zero constant.
2.  will be in GP, where k is any non-zero constant.

In the above two cases, the common ratio will be the same as the earlier.
3. If a GP of any even number of terms is given, then its common ratio will be the same as the ratio of the sum of all the even terms and the sum of all the odd terms.
Sum of n terms of a geometric progression
when r  1
Sn = na when r = 1
where n = number of terms, a = 1st term and d = common difference.

# Sum of infinite geometric progression

So far, we have done the summation of n terms of a GP. Now there is also a need of a separate expression for the sum of infinite GP. In case of AP, since either of the terms are always decreasing and going till −∝ or are always increasing and going till +. So, the summation of the infinite terms in AP will be either −∝ or +. However, the case is not the same in GP.

The need of a formula for infinite GP can be seen with the following example:

Find the sum of the series  2 + 1 + …till infinite terms.

In the above-written expression, the number of terms is not given, so we cannot find out the sum using the formula for finding the sum of n terms of a GP.

This can be further seen with the help of the following graphs:

If r is outside the range of –1 to 1, the terms of the series get bigger and bigger (even if they change the sign), and the series diverges. If r is within the range of –1 to 1, the terms get smaller and smaller (closer to 0) and the series converges.

Sum of Infinite GP = , where –1< r < 1.

Example-9
What is the sum of the following series: 1 + 2 + 4 +…till infinity.
Solution
It is very obvious that the sum is going to be +. This formula is applicable only for –1< r < 1.

Example-10
After striking the floor, a ball rebounds to 4/5th of the height from which it has fallen. What is the total distance that it travels before coming to rest if it is gently dropped from a height of 120 m?
Solution
The distance covered before the first rebound = 120 m
And then the ball bounces back to a height of 120 × and then falls from the same height.

Next time the ball will go up by 120 ×  × m and then it will fall from the same height.
So, the total distance covered
= 120 + 2 × 120 ×  + 2 × 120 ×  × +…∝.
= 120 + 2 × 120
= 120 + 240 = 1080 m

Alternatively, if the ball rebounds toth of the original height H, then the total distance covered = .

In the above example, the total distance covered
= 120 × = 1080 m

Example-11
On 1 January 2004, two new societies S1 and S2 are formed, each of n numbers. On the first day of each subsequent month, S1 adds b members while S2 multiples its current numbers by a constant factor r. Both the societies have the same number of members on 2 July 2004.

If b = 10.5n, what is the value of r? (CAT 2004)
1. 2.0
2. 1.9
3. 1.8
4. 1.7
Solution
There will be an increase of 6 times. The number of members in S1 will be in an AP.

On 2 July 2004, S1 will have n + 6b members
n + 6 × 10.5 n, = 64n

The number of members in S2 will be in a GP.

On 2 July 2004, the number of members in
S2 nr6

They are equal, hence 64 n = nr6
⇒ 64 = r6 ⇒ r = 2