Infinite Terms of a G.P

In geometric progression, the ratio of the successive terms should be constant. The constant value in the geometric progression is known as a common ratio. However, you can also say that the sum of G.P. is the sum of infinite terms. Moreover, geometrical progression can be infinite or finite. S infinite = a/1-r is the only formula when the geometrical progression series continues to infinity.

This formula is the first term of the series and is the common ratio between the terms in the series. However, there are three types of progression: geometric progression, arithmetic progression, and harmonic progression. But geometric progression is different from the rest. 

Sum of Infinite Terms of a G.P.

The formula a/1-r =Sn works when n is equal to infinity. However, n is the first term of a series and is in the common ratio in the G.P.

Where |r| < 1.

& r is not equal to 0.

Proving:

 Sn = a(1-rn) /1+r

= a-arn / 1-r

So the important result is  

= Sn= a/1-r – arn/1-r

rn=0 if n→∞

Using this formula, we can find out the infinity terms on G.P., but you should keep remembering that your module r is always less than one and not equal to 0.  

Finally,  

S∞ = a/1-r 

Another method to prove some of infinite GP

As we have learned above, with the formula of the sum of infinite terms of G.P., we will now discuss the standard ratio, which is ‘r’ where |r| < 1 and more significant than zero.

However, the sum of its infinite terms is when |r| < 1 and more potent than zero is:

Let us consider the first term of the series a, and r be the standard ratio. In addition, S is the number of series. 

S = a + ar + ar2 + ar3 + ……… Equation 1

On multiplying ‘r’ on both sides of equation 1, we get

rS = ar + ar2+ ar3 + ……….Equation 2 

On subtracting equation (2) from (1): 

S – rS = a

S (1 – r) = a 

Now, divide (1 – r) on both sides,  

S = a / (1 – r)

So, this is the formula for the sum of the infinity of G.P. when r is less than one and more significant than zero.

Problems Related to Sum of Infinite Terms of a G.P.

• Find the sum to identify the geometrical progression -5/4, 5/16, -5/64, 5/246…………..

The given series of geometrical progression is -5/4, 5/16, -5/64, 5/246………….. 

The first term a equals t -5/4, and the common ratio r is similar to -1/4. 

The sum of infinity will be S∞= a/1-r

Placing the value and formula, 

(5/4) /1-(1-¼) = -1

• Find the sum: S = 1/3 + 1 / (52) + 1 / (33) + 1/54 + 1/35+1/56 +…..………∞ 

S1 = 1/3+1/33+ 1/35 …….. 

S2 = 1/52+ 1/54 + 1/56 + …….  

As 

S1, a = 1/3 ( a is the first term of the first series) 

 r = 1/9 ( r is the common ratio of the first series)  

Hence, 

S1 = a/(1-r)

= (1/3)/(1-1/9)

= 3/8 

For S2,

a = 1/25 

r = 1/25

Hence, 

S1 = a/(1- r) 

= (1/25) / (1-1/25) 

= 1/24

On adding both series S1 and S2, we get 

S = S1 + S2 

3/8 + 1/24 

= 10/24 

= 5/12

So the final result is 5/12.

• Find the infinite sums of the G.P. series: 

(a) S = 1/3 + 1/9 + 1/27 +……….. 

(b) S = 1-1/2+1/4-1/8-……………

From the given series (a), 

S = 1/3 + 1/9 + 1/27 +……….. 

The first term of the series ‘a’ = 1/3 

The common ratio of the series ‘r’ = 1/3.

Where,

|r| = |1/3|

= 1/3 < 1.

The sum of infinite G.P. is,

S = a/(1-r)

= (1/3)/1 – (1/3)

= 1/2

From the given series (b),

S = 1-1/2+1/4-1/8-……………

We get, 

a = 1 (the first term of the series) 

r = -1/2 ( the common ratio of the series) 

Now

|r|= |1-½|

 = 1/2 < 1

The sum of infinite terms of G.P. is

S = a/(1-r)

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

= 2/3 

• Find the sums of infinity: 

(a) S = 1/3 + 1/9 + 1/27 +…….. 

(b) S = 1-1/2+1/4-1/8-…………..

From the series (a) 

a = 1/3 

r = 1/3 

As 

|r| = |1/3|

= 1/3 < 1. 

The infinite sum of G.P. is

S = a/(1-r)

= (1/3)/(1- (1/3)) 

= the important result is ½.

From the series (b),

 a = 1 and r = -1/2.

Here |r| = |1-1/2|

= 1/2 < 1.

The terms of the infinite sum of G.P. is

S = a/(1-r)

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

= so, the important result is 2/3

Conclusion

As you now learned, the geometric sequence is called geometric progression. However, the common ratio between the terms in the geometric progression series should be constant. Further, before solving the questions of the geometric progression, you should know about the derivation of the formula, which is related to the sum of infinity terms. All the questions of the sum of a geometric progression are related to the procedure, which is S infinity = a/(1-r). This is also a significant topic in mathematics. 

The subject also helps solve questions like trigonometric series, algebra series, number system series, etc. In conclusion, before solving the questions, first, solve the problem based on some of the infinitely many terms of a G.P. example and move towards the topic’s questions.