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Why a GP’s nth Term Has Exponent n−1

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Predicting What Comes Next: Exploring Sequences and Progressions · Geometric progressions · 3:35

You'll learn: Count repeated multiplications to derive tₙ=arⁿ⁻¹.

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Why a GP’s nth Term Has Exponent n−1

3:35

Playing

Why a GP’s nth Term Has Exponent n−1

You'll learn: Count repeated multiplications to derive tₙ=arⁿ⁻¹.

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Same topic

Geometric progressions

4

A GP Multiplies Where an AP Adds
3:54
A GP Multiplies Where an AP Adds
You'll learn: Compare equal differences with equal ratios using the two growth pictures.
Find the Common Ratio in Growing and Shrinking GPs
3:42
Find the Common Ratio in Growing and Shrinking GPs
You'll learn: Divide consecutive non-zero terms and derive term formulas.
Negative Ratios Can Make the Signs Alternate
3:53
Negative Ratios Can Make the Signs Alternate
You'll learn: Generate terms and verify a signed ratio.
Describe a GP with a Recursive Rule
3:51
Describe a GP with a Recursive Rule
You'll learn: Translate the explicit expression into a repeated-multiplier rule.

Next topic

GP term applications

2

Use a Known GP Term to Find a Later Term
3:31
Use a Known GP Term to Find a Later Term
You'll learn: Move a fixed number of positions by multiplying by the ratio that many times.
Which GP Term Equals a Given Number?
3:34
Which GP Term Equals a Given Number?
You'll learn: Match the target with a power of the common ratio.

Next topic

Fractals and geometric patterns

6

Build the Sierpiński Triangle One Removal at a Time
4:00
Build the Sierpiński Triangle One Removal at a Time
You'll learn: Join side midpoints and remove each central small triangle repeatedly.
Count the Small Triangles with a Power of Three
3:41
Count the Small Triangles with a Power of Three
You'll learn: Explain why each retained triangle produces three at the next stage.
More Triangles Can Occupy Less Total Area
3:41
More Triangles Can Occupy Less Total Area
You'll learn: Use the retained fraction to model the shrinking shaded area.
Plot AP and GP Behaviour Side by Side
3:55
Plot AP and GP Behaviour Side by Side
You'll learn: Compare constant additive change with multiplicative growth and decay.
Build and Count a Sierpiński Square Carpet
3:41
Build and Count a Sierpiński Square Carpet
You'll learn: Remove the central ninth in every retained square and count the remaining pieces.
Find the Shrinking Area of the Square Carpet
3:42
Find the Shrinking Area of the Square Carpet
You'll learn: Apply the retained fraction8/9 at every stage.

Next topic

Bouncing-ball applications

2

A Bouncing Ball’s Heights Form a Shrinking GP
3:50
A Bouncing Ball’s Heights Form a Shrinking GP
You'll learn: Generate bounce maxima and find the first bounce below a specified threshold.
Count Both Upward and Downward Travel Between Bounces
3:31
Count Both Upward and Downward Travel Between Bounces
You'll learn: Distinguish rebound height from total vertical distance travelled.

Next topic

Chapter challenge problems

1

A Shifted Recursive Sequence Can Reveal a GP
3:28
A Shifted Recursive Sequence Can Reveal a GP
You'll learn: Transform the recurrence into a geometric rule and identify a target term.