Question
Derivation of the formula for the sum of a geometric sequence
Derivation of the formula for the sum of a geometric sequence
The formula for sum of a geometric sequence can be derived from the formula for general terms of a geometric sequence.
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}
The sum of a geometric sequence is
S_n=a_1+a_2+\dots+a_{n-1}+ a_n
Using the formula for the n^{th} term of a geometric sequence,
a_n = a_1r^{n-1}
Multiply both sides of the equation by r
Subtract (2) from (1)
S_n-rS_n= a_1-a_1r^n
(1-r)S_n = a_1(1-r^n)
S_n=\dfrac{ a_1(1-r^n)}{1-r}