MYP Extended Mathematics Topic Geometric Sequences Revision Notes

MYP Extended Mathematics Topic Geometric Sequences Revision Notes | RevisionDojo

Recognize Geometric Sequences by a Constant Ratio

Definition

Geometric sequence

A sequence in which the ratio between consecutive terms is constant (each term is found by multiplying the previous term by the same number).

A sequence is an ordered list of numbers (called terms) such as $6, 42, 294, \ldots$.
In many practical settings, the terms change by the same multiplier each step, rather than by the same amount.
That is exactly the idea of a geometric sequence.
A sequence $u_n$ is geometric if there is a constant number $r$ such that $$\frac{u_{n+1}}{u_n}=r$$ for every term where this division is defined.

Note

Different books use different letters.
You may see the first term written as $a$ and the ratio as $q$.
We use $u_1$ for the first term and $r$ for the common ratio.

Understand the Common Ratio and How It Controls Behavior

Definition

Common ratio

The constant multiplier $r$ in a geometric sequence, found by dividing a term by the previous term: $r=\frac{u_{n+1}}{u_n}$.

Example

If a geometric sequence begins $4, 12, 36, 108, \ldots$, then $$r=\frac{12}{4}=\frac{36}{12}=\frac{108}{36}=3,$$ so every term is the previous term multiplied by 3.

Different Values of $r$ Produce Different Patterns

The value of the common ratio $r$ determines the long-term behavior.

$r>1$: if $u_1>0$, the terms increase rapidly (geometric growth).
$0<r<1$: if $u_1>0$, the terms decrease toward 0 (geometric decay).
$r=1$: all terms are equal (constant sequence).
$r<0$: the terms alternate in sign, because multiplying by a negative number flips the sign each step.

Example

The sequence $5, -10, 20, -40, 80, \ldots$ is geometric with $$r=\frac{-10}{5}=-2$$
Because $r$ is negative, the signs alternate: $+, -, +, -, +, \ldots$.

Note

When $r$ is negative, the terms of a geometric sequence alternate in sign.
The sizes still grow if $|r|>1$.

Common Mistake

The constant sequence $3, 3, 3, 3, \ldots$ is geometric (with $r=1$).
However, the all-zero sequence $0, 0, 0, 0, \ldots$ is usually not treated as geometric because ratios like $\frac{0}{0}$ are undefined, so the "constant ratio" test breaks down.

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MYP Extended Mathematics Geometric Sequences Notes

Recognize Geometric Sequences by a Constant Ratio

Understand the Common Ratio and How It Controls Behavior

Different Values of $r$ Produce Different Patterns

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Recognize Geometric Sequences by a Constant Ratio

Understand the Common Ratio and How It Controls Behavior

Different Values of $r$ Produce Different Patterns

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Advanced Number and Bounds3 subtopics

Inequalities and Modelling1 subtopic

Function Transformations4 subtopics

Advanced Probability1 subtopic

Geometric Transformations1 subtopic

Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

Covariance and Correlation3 subtopics

Logarithms and Non-Linear Inequalities3 subtopics

Graph Theory and Vectors4 subtopics

MYP Extended Mathematics Geometric Sequences Notes

Advanced Number and Bounds3 subtopics

Inequalities and Modelling1 subtopic

Function Transformations4 subtopics

Advanced Probability1 subtopic

Geometric Transformations1 subtopic

Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

Covariance and Correlation3 subtopics

Logarithms and Non-Linear Inequalities3 subtopics

Graph Theory and Vectors4 subtopics

Recognize Geometric Sequences by a Constant Ratio

Understand the Common Ratio and How It Controls Behavior

Different Values of $r$ Produce Different Patterns

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Recognize Geometric Sequences by a Constant Ratio

Understand the Common Ratio and How It Controls Behavior

Different Values of $r$ Produce Different Patterns

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Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

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