A geometric series is formed by multiplying each term by the same constant ratio \(r\).
If the first term is \(a\), the terms are:
\[
a,\quad ar,\quad ar^2,\quad ar^3,\quad \ldots
\]
1. General term
The \(n\)-th term is:
\[
a_n=ar^{n-1}
\]
This formula starts with \(a_1=a\).
2. Finite geometric sum
The sum of the first \(N\) terms is:
\[
S_N=a+ar+ar^2+\cdots+ar^{N-1}
\]
When \(r\ne1\), the finite sum is:
\[
S_N=a\frac{1-r^N}{1-r}
\]
When \(r=1\), all terms are equal to \(a\), so:
\[
S_N=aN
\]
3. Derivation of the finite formula
Start with:
\[
S_N=a+ar+ar^2+\cdots+ar^{N-1}
\]
Multiply by \(r\):
\[
rS_N=ar+ar^2+ar^3+\cdots+ar^N
\]
Subtract:
\[
S_N-rS_N=a-ar^N
\]
\[
S_N(1-r)=a(1-r^N)
\]
\[
S_N=a\frac{1-r^N}{1-r}
\]
4. Infinite geometric series
An infinite geometric series is:
\[
S_\infty=a+ar+ar^2+ar^3+\cdots
\]
It converges only when:
\[
|r|\lt1
\]
In that case:
\[
S_\infty=\frac{a}{1-r}
\]
If \(|r|\ge1\), the infinite geometric series diverges, unless \(a=0\).
5. Example: finite sum
Find the sum of:
\[
3+6+12+\cdots
\]
for \(8\) terms.
\[
a=3,\qquad r=2,\qquad N=8
\]
\[
S_8=3\frac{1-2^8}{1-2}
\]
\[
S_8=3\frac{1-256}{-1}
\]
\[
S_8=765
\]
6. Example: infinite sum
Consider:
\[
5+2.5+1.25+\cdots
\]
Here:
\[
a=5,\qquad r=0.5
\]
Since \(|0.5|\lt1\), the infinite sum exists:
\[
S_\infty=\frac{5}{1-0.5}=10
\]
7. Alternating geometric series
If \(r\) is negative, the terms change signs:
\[
4-2+1-\frac12+\cdots
\]
Here \(a=4\) and \(r=-0.5\). Since \(|r|\lt1\), the infinite series converges:
\[
S_\infty=\frac{4}{1-(-0.5)}=\frac{4}{1.5}=\frac{8}{3}
\]
8. Graph interpretation
The bars show the individual terms \(a_n\). The line shows the partial sums \(S_n\).
When \(|r|\lt1\), the partial-sum line approaches a horizontal limit.
If \(r\gt1\), the bars usually grow larger and the partial sums increase or decrease without bound.
If \(r\le-1\), the partial sums may oscillate with increasing size.
9. Summary table
| Situation |
Formula |
Condition |
| General term |
\(a_n=ar^{n-1}\) |
Any real \(a,r\) |
| Finite sum |
\(S_N=a\dfrac{1-r^N}{1-r}\) |
\(r\ne1\) |
| Finite sum special case |
\(S_N=aN\) |
\(r=1\) |
| Infinite sum |
\(S_\infty=\dfrac{a}{1-r}\) |
\(|r|\lt1\) |
| Infinite divergence |
No finite sum |
\(|r|\ge1\), usually |
10. Common mistakes
- Do not use the infinite formula unless \(|r|\lt1\).
- Do not forget the special case \(r=1\).
- For a finite sum, the formula works even when the infinite series diverges.
- The number \(N\) is the number of terms, not always the last exponent.
- If the first term is \(a\), the last term in \(S_N\) is \(ar^{N-1}\).
