Consider the following three sequences:
\( 2,\dfrac{2}{3},\dfrac{2}{9},\dfrac{2}{27},\dfrac{2}{81},\cdots\)
\( 1,\dfrac{1}{2},\dfrac{1}{8},\dfrac{1}{16},\dfrac{1}{32},\cdots\)
\( 3,-1,\dfrac{1}{3},-\dfrac{1}{9},\dfrac{1}{27},\cdots\)
What each of these sequences has in common is that ratio of two successive terms is the same constant for each particular sequence. That constant is called the common ratio of the sequence. For the first sequence, the common ratio is one-third, since each term beginning with the second term is exactly one-third of the preceding term. For the second sequence, the common ratio is one-half and for the third sequence, it is negative one-third. Sequences defined in this manner by a common ratio are called geometric sequences.
Geometric sequences can be defined in terms of their initial term and the common ratio, \(r\).
\(c_2=c_1r\)
\(c_3=c_2r=(c_1r)r=c_1r^2\)
\(c_4=c_3r=(c_1r^2)r=c_1r^3\)
\(c_5=c_4r=(c_1r^3)r=c_1r^4\), et cetera.
So we see that the general formula for the \(n\)-th term is:
\[c_n=c_1r^{n-1}\]
Find the general formula for each of the three geometric sequences above.
A geometric series is a sum of the terms of a geometric sequence.
Suppose \(S\) denote the sum of the first 10 terms of the first sequence above. Then
\[S=\sum_{n=1}^{10}2\left(\frac{1}{3}\right)^{n-1}=2\left(\frac{1}{3}\right)^{0}+2\left(\dfrac{1}{3}\right)^1+2\left(\dfrac{1}{3}\right)^2+\cdots+2\left(\frac{1}{3}\right)^{9}\]
If we multiple both sides by the common difference, we get
\[\frac{1}{3}S=2\left(\frac{1}{3}\right)^{1}+2\left(\dfrac{1}{3}\right)^2+2\left(\dfrac{1}{3}\right)^3+\cdots+2\left(\frac{1}{3}\right)^{10}\]If we subtract the second equation from the first, all the terms in common between the two sums subtract away leaving the first term of the first equation minus the last term of the second equation, giving the result
\[\frac{2}{3}S=2\left(\frac{1}{3}\right)^{0}-2\left(\frac{1}{3}\right)^{10}\]
Multiplying both sides by \(\dfrac{3}{2}\) gives
\[S=3-3\left(\frac{1}{3}\right)^{10}\approx 2.999949\]Find the sums of the first ten terms of the second and third geometric series at the beginning of this section.
The sum of the first \(n\) terms of a finite geometric series is:
\[S_n=c_1+c_1r+c_1r^2+\cdots+c_1r^{n-1}=c_1\left(\frac{1-r^n}{1-r}\right)\]
If the absolute value of \(r\) is smaller than one, then \(r^n\) becomes vanishingly small as \(n\) becomes large. Thus, the sum of the infinite geometric series converges when \(| r | < 1\).
\[S_n=c_1+c_1r+c_1r^2+\cdots=\frac{c_1}{1-r}\]
Find the sum of the infinite geometric series: \(\displaystyle\sum_{n=1}^\infty4\left(-\frac{3}{4}\right)^{n-1}\)