Algebra

Sequences

Explore constant steps, square and triangle numbers, and Fibonacci.

Predict, test and explain

Step 1 of 3

Predict

A pattern starts with 2 tiles and adds 3 each stage. How many tiles are at stage 4?

T4=11T_{4}=11
stage 1\text{stage }1
22
stage 2\text{stage }2
55
stage 3\text{stage }3
88
stage 4\text{stage }4
1111

Scroll the diagram sideways to see every label.

Tile counts2,  5,  8,  112,\;5,\;8,\;11

InvestigatePredict the next stage before revealing it.

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Find the nth-term rule for this arithmetic sequence; stage one is the first term.

5, 8, 11, 14, …5,\ 8,\ 11,\ 14,\ \ldots

Hint

The common difference is three. Adjust the constant so stage one gives five.

Reveal answer

Multiply the stage number by three, then add two.

un=3n+2(n=1,2,3,…)u_n=3n+2\quad(n=1,2,3,\ldots)

Build an explanation

Hint 1

Compare consecutive stages and count what is added.

Hint 2

A constant difference gives a linear rule; stage 1 is the first term.

Hint 3

For first term a and difference d, term n is a + (n − 1)d.

Worked example

This example uses fixed values, separate from the diagram controls.

A pattern starts with 2 tiles and adds 3 each stage. Find stage 4 and the nth-term rule.

  1. Stage 4 has 2 + (4 − 1) × 3 = 11 tiles.
  2. Expand 2 + (n − 1) × 3.

Answer: Stage 4: 11 tiles; nth term: 3n − 1.

Connect this idea

Watch out: The common difference alone is not the full nth-term rule; include the starting offset.