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Infinite Geometric Series Formula

An infinite geometric series is the sum of an infinite geometric sequence. This series would have no last term. The general form of the infinite geometric series is $a_{1}+a_{1}r+a_{1}r^{2}+a_{1}r^{3}+….$ ,where $a_{1}$r is the first term and r is the common ratio.

The Infinite Geometric Series Formula is given as,

\[\large a_{1}+a_{1}r+a_{1}r^{2}+a_{1}r^{3}+….+a_{1}r^{n-1}\]

The formula for the resultant sum of the Infinite Geometric Series is,

\[\large S_{\infty}=\frac{a_{1}}{1-r};\left|r\right|<1\]

The following table shows several geometric series with different common ratios:

Common ratio, r Start term, a Example series
10 4 4 + 40 + 400 + 4000 + 40,000 + ···
1/3 9 9 + 3 + 1 + 1/3 + 1/9 + ···
1/10 7 7 + 0.7 + 0.07 + 0.007 + 0.0007 + ···
1 3 3 + 3 + 3 + 3 + 3 + ···
−1/2 1 1 − 1/2 + 1/4 − 1/8 + 1/16 − 1/32 + ···
–1 3 3 − 3 + 3 − 3 + 3 − ···

Solved Examples

Question: Find the sum of the geometric series 125 + 25 + 5 + 1 +…… ?

Solution:

The series is, 125 + 25 + 5 + 1 +…..

a1 = 125

r = $\frac{25}{125}$ = $\frac{1}{5}$

The formula for the resultant sum of the Infinite Geometric Series is,

S∞ = $\frac{a_{1}}{1 – r}$

S∞ = $\frac{125}{1 – \frac{1}{5}}$

S∞ = $\frac{125}{\frac{4}{5}}$

S∞ = $\frac{625}{4}$