Here you will learn sum of gp to infinity (sum of infinite gp) and its proof with examples.
Let’s begin –
Sum of GP to Infinity (Sum of Infinite GP)
The sum of an infinite GP with first term a and common ratio r(-1 < r < 1 i.e. , | r | < 1) is
S = \(a\over 1-r\)
Also Read : Sum of GP Series Formula | Properties of GP
Proof : Consider an infinite GP with first term a and common ratio r, where -1 < r < 1 i.e. , | r | < 1. The sum of n terms of this GP is given by
\(S_n\) = a\({1-r^n}\over {1-r}\) = \(a\over {1-r}\) – \(ar^n\over {1-r}\) ……..(i)
Since -1 < r < 1, therefore \(r^n\) decreases as n increases and \(r^n\) tends to zero as n tends to infinity i.e. \(r^n\) \(\rightarrow\) 0 as n \(\rightarrow\) \(\infty\).
\(\therefore\) \(ar^n\over {1-r}\) \(\rightarrow\) 0 as n \(\rightarrow\) \(\infty\).
Hence from (i), the sum of an infinite GP is given by
S = \(lim_{n \to \infty}\) \(S_n\) = \(lim_{n \to \infty}\) ( \(a\over {1-r}\) – \(ar^n\over {1-r}\) ) = \(a\over 1-r\), if | r | < 1
Note : If r \(\ge\) 1, then the sum of an infinite GP tends to infinity.
Example : Find the sum to infinity of the GP \(-5\over 4\), \(5\over 16\), \(-5\over 64\), ……
Solution : The given GP has the first term a = -5/4 and the common ratio r = -1/4
Also | r | < 1.
Hence the sum of an infinite GP is given by S = \(a\over {1-r}\)
S = \(-5/4\over {1-(-1/4)}\) = -1
Example : The sum of an infinite GP is 57 and the sum of their cubes is 9747, find the GP.
Solution : Let a be the first term and r be the common ratio of the GP. Then
Sum = 57 \(\implies\) \(a\over 1-r\) = 57 …….(i)
Sum of the cubes = 9747
\(\implies\) \(a^3\) + \(a^3r^3\) + \(a^3r^6\) + ….. = 9747
\(\implies\) \(a^3\over {1 – r^3}\) = 9747 ……..(ii)
Dividing the cube of (i) by (ii), we get
\(a^3\over {(1-r)}^3\) . \((1-r^3)\over a^3\) = \({(57)}^3\over 9747\)
\(\implies\) \(1 – r^3\over {(1 – r)}^3\) = 19
= \(1+r+r^2\over {(1-r)}^2\)
= \(18r^2\) – 39r + 18 = 0
\(\implies\) (3r-2)(6r-9) = 0
\(\implies\) r = 2/3 or r = 3/2
Hence r = 2/3 [ \(\because\) r \(\ne\) 3/2, because -1 < r < 1 for an infinite GP]
Putting r = 2/3 in equation (i), we get
\(a\over {(1-(2/3))}\) = 57 \(\implies\) a = 19
Hence, the GP is 19, 38/3, 76/9, …….
