UPSKILL MATH PLUS
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Learn moreIn this section, we will learn about geometric series.
Geometric series:
A geometric series is a series with all of its terms in a G.P.
Let us now try to find the sum of the n terms in a G.P.
Let the terms in G.P be a, ar, ar^2, ar^3,…,ar^{n-1}.
S_n = a + ar + ar^2 + ar^3 +...+ ar^{n - 1} \longrightarrow (1)
On multiplying by r on both the sides, we get:
rS_n = ar + ar^2 + ar^3 + ar^4 +...+ ar^{n} \longrightarrow (2)
On subtracting (2) from (1), we get:
rS_n - S_n = ar^{n} - a
S_n(r - 1) = a(r^n -1)
S_n = \frac{a(r^n -1)}{r - 1}
Sum of n terms in a series when r = 1:
S_n = a + ar + ar^2 + ar^3 +...+ ar^{n - 1}
S_n = a + a(1) + a(1)^2 + a(1)^3 +...+ a(1)^{n - 1}
S_n = a + a + a + a +...+ a
S_n = na
Sum of infinite terms in a series:
\text{Sum of infinite terms in a series} = a + ar + ar^2 + ar^3 +...
\text{Sum of infinite terms in a series} = \frac{a}{1 - r}, -1 < r < 1
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