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The nth Triangular Number Is a Sum

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Predicting What Comes Next: Exploring Sequences and Progressions · Sums of natural numbers and triangular numbers · 3:31

You'll learn: Connect the original triangle arrays to the closed term formula.

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The nth Triangular Number Is a Sum

3:31

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The nth Triangular Number Is a Sum

You'll learn: Connect the original triangle arrays to the closed term formula.

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Same topic

Sums of natural numbers and triangular numbers

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Pair a Sum with Its Reverse
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Pair a Sum with Its Reverse
You'll learn: Add1–10 in opposite directions so every column has the same total.
Derive the Sum of the First n Natural Numbers
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Derive the Sum of the First n Natural Numbers
You'll learn: Generalise the reversed-sum proof and connect it to a rectangular dot array.
Find a Sum That Starts Somewhere Other Than One
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Find a Sum That Starts Somewhere Other Than One
You'll learn: Subtract two initial sums to retain exactly the intended interval.

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Natural-number sums and AP applications

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Count and Add the Two-Digit Multiples of Three
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Count and Add the Two-Digit Multiples of Three
You'll learn: Identify the first and last terms, count the steps and add the required multiples.

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Geometric progressions

5

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.
Why a GP’s nth Term Has Exponent n−1
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Why a GP’s nth Term Has Exponent n−1
You'll learn: Count repeated multiplications to derive tₙ=arⁿ⁻¹.
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
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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?
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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

4

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
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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
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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
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Plot AP and GP Behaviour Side by Side
You'll learn: Compare constant additive change with multiplicative growth and decay.