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In 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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