Sum of Infinite Terms of GP Formula Meaning and Examples

Sum of Infinite Terms of GP Formula: GP, formally known as the Geometric Progression, is one of the most important concepts in mathematics which finds its applications in many different branches ranging from physics to finance. GP is one of the progression series. Other progression series are Arithmetic Progression (AP) and Harmonic Progression (HP). One such formula related to the Geometric Progression is the sum of infinite terms of GP formula. Let us explore various aspects of this formula and understand its different concepts.

Sum of Infinite Terms of GP Formula Meaning

The sum of Infinite terms of GP means the addition of the terms present in the geometric progression from the first term to infinite values. Before understanding the sum of infinite terms of GP, first, let us get acquainted with the term GP (Geometric Progression). Geometric Progression is basically a series of numbers in which the succeeding term is more or less than the preceding term by a certain multiplicative factor. In other words, the ratio between two consecutive terms in a Geometric Progression is always constant. The Geometric progression can be finite or infinite. In finite GP, the number of terms can be counted while in the infinite GP, the number of terms in a GP is infinity, i.e., it cannot be counted.

Sum of Infinite Terms of GP Formula Example

Before looking into the example of the sum of the infinite terms of a GP, first, let us go through the example of the geometric progression series. The formula for the GP series is given below:

a, ar, ar², ar³, and so on

Here the common ratio is “r” in each case which can be obtained by taking the ratio of any consecutive terms. The common is always constant for a GP.

That is, ar/a = ar²/ar = ar³/ar²

The example of some GP are:

4, 8, 16, 32, 64, 128: It is a finite GP

9, 27, 81, 243,………….. : It is an infinite GP as it is continuing to infinity as shows by the dots

Now, the term, “sum of infinite GP”, means the total value of the GP when the terms of an infinite GP are added.

that is, 9 + 27 + 81 + 243 + ……………..

It is interesting to note that

Sum of Infinite Terms of GP Formula

After understanding the meaning of Infinite GP, let us now have a look at the formula to find the sum of infinite terms of a GP. The formula for the sum of infinite terms of a GP can be found by modifying the formula for the sum of finite terms of a GP. The formula for the sum of finite terms of a GP is given by:

S = a(rⁿ – 1) / (r – 1)  for “r”>=1

or S = a(1 – rⁿ) / (1 – r), when “r”<1

where, r = common ratio

a = first term of the series

n = number of terms in a GP

S = sum of n terms

So from here, we can conclude that the formula for the sum of infinite terms of GP will have two different forms that will depend on the common ratio.

Sum of Infinite Terms of GP Formula when r is greater than 1

In such infinite GP, the common ratio is greater than 1. For example the series: 2, 4, 8, 16, 32, 64, 128, 256, …….. is an infinite GP with common ratio of 2. As we can observe from the above-given infinite series, the values of each successive terms gets bigger and bigger. In this case, after a certain point in the series, the terms of the series expand quickly and take on very large values. Consequently, the series diverges, failing to converge to a finite value, and the sum of a GP with r > 1 cannot be obtained. So we can say that, the sum of infinite terms of a GP formula does not exist when r > 1.

Sum of Infinite Terms of GP Formula when r is equal to 1

In this type of series, each successive term is equal to the previous term. For example, 8, 8, 8, 8, 8, 8, 8……………… The behavior of the GP series with r = 1 is same as that of r > 1. So, it can be said that the sum of infinite terms of a GP cannot be found if r = 1.

Sum of Infinite Terms of GP Formula when r is less than 1

The formula for this series exist unlike the other two series mentioned above. In this series, each successive term is less than the previous term. The example of such infinite series are given below:

1/2, 1/4, 1/8, 1/16, 1/32,…………..

9, 3, 1, 1/3, 1/9, 1/27, 1/81, 1/243,…………..

As we can see from the above series that each successive term of the GP is less than the previous term. In this case, we can see the series converging towards smaller and smaller values with each successive term. So we can find the sum of such series using the formula. So the sum of series like 9 + 3 + 1 + 1/3 + 1/9 + 1/27 + 1/81 + 1/243 + ………… can be found using this formula.

The sum of infinite terms of GP Formula for r < 1 is given by:

S =  a/(1-r)

where S = sum

a = first term of the series

r = common ratio

Contrary to popular belief, an infinite geometric progression (GP) has a finite sum when the common ratio’s absolute value is less than 1, contrary to the usual notion that the sum of an infinite number of terms equals infinity.

Sum of Infinite Terms of GP Formula Proof

We can find out the proof for the sum of infinite terms of GP formula by using simple mathematical calculations. The proof for the same is given below.

Let the infinite GP series be a, ar, ar², ar³,………

S = a + ar + ar² + ar³ + …………………. (i)

where, S = sum of the series

multiplying equation (i) by r on both sides

rS = ar + ar² + ar³ + ar⁴ +……………………………. (ii)

subtracting the equation (ii) from equation (i), we get:

S – rS = (a + ar + ar² + ar³ + ………………….) – (ar + ar² + ar³ + ar⁴ +…………………………….)

S (1-r) = a

or, S = a/(1-r)

Hence proved

Sum of Infinite Terms of GP Formula Example

Some of the solved examples on sum of infinite terms of GP formula is given below. By going through these questions, students will have a better understanding on how to use the sum of infinite terms of GP formula.

Example 1: What will be the sum of an infinite GP whose first term is 6 and the common ratio is 1/2.

Solution: As the GP is infinite and given,

First term (a) = 6

common ration (r) = 1/2

Using the sum of infinite terms of GP formula S = a/(1-r)

S = 6/(1-1/2)

S= 6/(1/2)

S = 12

Example 2: Consider a GP with a = 2 and its sum = 3. The GP is given to be infinite. Calculate its common ratio.

Solution: Given

a = 2

S = 3

Using the sum of infinite terms of GP formula S = a/(1-r)

3 = 2/(1-r)

or, (1-r) = 2/3

r = 1-(2/3)

Or, r = 1/3

Hence the common ratio is 1/3

Example 3: Find out the sum of the series of the series 1/2 + 1/4 + 1/8 + 1/16 + ….. up to infinity.

Solution: The series given is 1/2 + 1/4 + 1/8 + 1/16 + …………..

First term (a) = 1/2

Common ratio (r) = 1/2

Using the sum of infinite terms of GP formula given by

S = a/(1-r)

S = (1/2)/(1-1/2)

S = (1/2)/(1/2)

S= 1

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