Geometric Sequences and Series
Arithmetic sequences differ by a common difference $d$ each term. Geometric sequences are similar, but differ by a common ratio $r$ each term instead, where the next term is the previous term multiplied by $r$.
Definition and Basic Properties
A geometric sequence is defined by its first term $u_1$ and common ratio $r$. The general form of a geometric sequence is:
$$u_1, u_1r, u_1r^2, u_1r^3, ..., u_1r^{n-1}$$
where $n$ is the position of the term in the sequence.
ExampleIf we have a sequence 2, 6, 18, 54, ..., we can identify that:
- The first term $a = 2$
- The common ratio $r = \frac{6}{2} = 3$ This sequence can be written as $2, 2(3), 2(3^2), 2(3^3), ...$
The common ratio of a geometric sequence can be between 0 and 1, in which case the terms in the sequence decrease.
For example, the first five terms of a geometric series with $u_1 = 8$ and $r = \frac12$ would be $8, 4, 2, 1, \frac12$.
The common ratio can also be negative, in which case the terms alternate sign.
For example, the first five terms of a geometric series with $u_1 = 2$ and $r = -3$ would be $2, -6, 18, -54, 162$.
The nth Term Formula
The formula for the nth term of a geometric sequence is:
$$u_n = u_1r^{n-1}$$
where $a_n$ is the nth term, $u_1$ is the first term, $r$ is the common ratio, and $n$ is the position of the term.
TipTo find the common ratio $r$, divide any term by the previous term in the sequence. This should give the same result for any pair of consecutive terms.
Geometric Series
A geometric series is the sum of $n$ terms of a geometric sequence. The sum of the first $n$ terms of a geometric sequence is given by the formula:
$$S_n = \frac{u_1(1-r^n)}{1-r} = \frac{u_1(r^n - 1)}{r-1} \, \text{ for } r \neq 1$$
where $S_n$ is the sum of the first $n$ terms, $u_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.
NoteThe formula for $r \neq 1$ can be derived using the concept of difference of geometric series. It's a powerful tool for quickly calculating sums that would otherwise be tedious to compute manually.
Sigma Notation
Sigma notation can represent geometric sequences as well as arithmetic. For geometric sequences, it's written as:
$$\sum_{k=1}^n u_1r^{k-1}$$
This notation means sum all terms from $k=1$ to $n$, where each term is given by $u_1r^{k-1}$.
ExampleThe sum of the first 5 terms of the sequence 2, 6, 18, 54, ... can be written as:
$$\sum_{k=1}^5 2(3^{k-1})$$
This is equivalent to $2 + 6 + 18 + 54 + 162 = 242$
NoteIn IB examinations where calculators are allowed, students are expected to identify the first term and common ratio from given data or graphs.
