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Sum of an infinite Geometric Progression

The sum of an infinite Geometric Progression whose first term 'a' and common ratio 'r' (-1 < r < 1 i.e., |r| < 1) is

S = a1βˆ’r

Proof:

A series of the form a + ar + ar2 + ...... + arn + ............... ∞ is called an infinite geometric series.

Let us consider an infinite Geometric Progression with first term a and common ratio r, where -1 < r < 1 i.e., |r| < 1. Therefore, the sum of n terms of this Geometric Progression in given by

Sn = a(1βˆ’rn1βˆ’r) = a1βˆ’r - arn1βˆ’r ........................ (i)

Since - 1< r < 1, therefore rn  decreases as n increases and r^n tends to zero an n tends to infinity i.e., rn β†’ 0 as n β†’ ∞.

Therefore,

arn1βˆ’r β†’ 0 as n β†’ ∞.

Hence, from (i), the sum of an infinite Geometric Progression ig given by

S = limxβ†’0 Sn  = limxβ†’βˆž(a1βˆ’rβˆ’ar21βˆ’r) = a1βˆ’r if |r| < 1


Note: (i) If an infinite series has a sum, the series is said to be convergent. On the contrary, an infinite series is said to be divergent it it has no sum. The infinite geometric series a + ar + ar2 + ...... + arn + ............... ∞ has a sum when -1 < r < 1; so it is convergent when -1 < r < 1. But it is divergent when r > 1 or, r < -1.

(ii) If r β‰₯ 1, then the sum of an infinite Geometric Progression tens to infinity.


Solved examples to find the sum to infinity of the Geometric Progression:

1. Find the sum to infinity of the Geometric Progression

-54, 516, -564, 5256, .........

Solution:

The given Geometric Progression is -54, 516, -564, 5256, .........

It has first term a = -54 and the common ratio r = -14. Also, |r| < 1.

Therefore, the sum to infinity is given by

S = a1βˆ’r = 541βˆ’(βˆ’14)  = -1


2. Express the recurring decimals as rational number: 3Λ™6

Solution:

3Λ™6 = 0.3636363636............... ∞

= 0.36 + 0.0036 + 0.000036 + 0.00000036 + .................. ∞

= 36102 + 36104 + 36106 + 36108 + .................. ∞, which is an infinite geometric series whose first term = 36102 and common ratio = 1102 < 1.

= 361021βˆ’1102, [Using the formula S = a1βˆ’r]

= 361001βˆ’1100

= 36100100βˆ’1100

= 3610099100

= 36100 Γ— 10099

= 411

● Geometric Progression




11 and 12 Grade Math 

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