IB Mathematics Analysis and Approaches Topic SL 1.3—geometric Sequences and Series Revision Notes (SL/HL)

IB Mathematics Analysis and Approaches Topic SL 1.3—geometric Sequences and Series Revision Notes (SL/HL) | RevisionDojo

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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

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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

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Number and Algebra16 subtopics

Functions16 subtopics

Geometry & Trigonometry18 subtopics

Statistics & Probability14 subtopics

Calculus19 subtopics

IB Mathematics Analysis and Approaches SL 1.3—geometric Sequences and Series Notes

Geometric Sequences and Series

Definition and Basic Properties

The nth Term Formula

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Geometric Sequences and Series

Definition and Basic Properties

The nth Term Formula

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Number and Algebra16 subtopics

Functions16 subtopics

Geometry & Trigonometry18 subtopics

Statistics & Probability14 subtopics

Calculus19 subtopics