Subscribe to our βΆοΈ YouTube channel π΄ for the latest videos, updates, and tips.
Home | About Us | Contact Us | Privacy | Math Blog
The sum of an infinite Geometric Progression whose first term 'a' and common ratio 'r' (-1 < r < 1 i.e., |r| < 1) is
S = a1βr
Proof:
A series of the form a + ar + ar2 + ...... + arn + ............... β 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(1βrn1βr) = a1βr - arn1βr ........................ (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 β.
Therefore,
arn1βr β 0 as n β β.
Hence, from (i), the sum of an infinite Geometric Progression ig given by
S = limxβ0 Sn = limxββ(a1βrβar21βr) = a1βr 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 + ............... β 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 = a1βr = 541β(β14) = -1
2. Express the recurring decimals as rational number: 3Λ6
Solution:
3Λ6 = 0.3636363636............... β
= 0.36 + 0.0036 + 0.000036 + 0.00000036 + .................. β
= 36102 + 36104 + 36106 + 36108 + .................. β, which is an infinite geometric series whose first term = 36102 and common ratio = 1102 < 1.
= 361021β1102, [Using the formula S = a1βr]
= 361001β1100
= 36100100β1100
= 3610099100
= 36100 Γ 10099
= 411
β Geometric Progression
11 and 12 Grade Math
From Sum of an infinite Geometric Progression to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Jul 29, 25 12:59 AM
Jul 28, 25 01:52 PM
Jul 25, 25 03:15 AM
Jul 24, 25 03:46 PM
Jul 23, 25 11:37 AM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.