Question

Derivation of sum of a geometric sequence

Steven Zheng Steven Zheng posted 2 months ago


Answer

Using the formula for the n^{th} term of a geometric sequence,

a_n = a_1r^{n-1}

in which a_1 is the first tirm, r is the common ratio.

a_1 = a_1r^0

a_2 = a_1r^1

a_3 = a_1r^2

\dots

\begin{aligned} S_n&=a_1+a_2+\dots+ a_n \\ &=a_1(1+r^1+r^2+\dots+r^{n-1}) \\ &= \dfrac{a_1(r-1)(1+r^1+r^2+\dots+r^{n-1})}{r-1} \\ &= \dfrac{a_1(r+r^2+\dots+r^n-1 -r-r^2-\dots-r^{n-1})}{r-1} \\ &= \dfrac{a_1(1-r^n)}{1-r} \end{aligned}

Steven Zheng Steven Zheng posted 2 months ago

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