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.
Example
- If 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 $2, 2(3), 2(3^2), 2(3^3), ...$
Note
- 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
- $n$ is the position of the term
Hint
- To 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: $$S_n = \frac{u_1(1-r^n)}{1-r} \text{ for } r \neq 1$$ $$S_n = nu_1 \text{ for } r = 1$$ where
- $S_n$ is the sum of the first $n$ terms
- $a$ is the first term
- $r$ is the common ratio
- $n$ is the number of terms
Note
- The formula for $r \neq 1$ can be derived using the concept of difference of geometric series.
- It is 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 $u_1r^{k-1}$.
Example
- The 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 $2 + 6 + 18 + 54 + 162 = 242$
Exam technique
In IB examinations where calculators are allowed, students are expected to identify the first term and common ratio from given data or graphs.
