Sequences from a rule

A sequence starts `2, 9, 16, 23, ...`. Write the term-to-term rule in words, then find the next two terms.

Art hook Draw a horizontal number line; place a dot at 2, then hop 7 units at a time, drawing an arc for each jump. Each new arc gets the next hue on a colour wheel, so equal jumps become equally-spaced coloured arcs marching rightward.

A pattern of squares grows so that the number of squares is `3, 5, 7, 9, ...`. First work out the position-to-term rule (the rule that gives a term straight from its position number, position 1 -> 3, position 2 -> 5, ...). Then use your rule to find how many squares are in the 20th pattern WITHOUT listing all the earlier patterns.

Art hook For each position p, draw a row of (2p+1) dots centred on a vertical axis — a growing symmetric arrow/triangle. Stack rows p = 1..20 to see the staircase, with the centre column highlighted.

Follow this calculation program on the positions `1, 2, 3, 4`: 'take the position, multiply by 3, then subtract 2'. Write the four terms you get.

Art hook Show a machine with an input dot (position) entering, a '×3' gear and a '−2' gear, and an output dot dropping onto a number line. Feed 1,2,3,4 in and watch the outputs land at evenly spaced points 3 apart.

Here is a sequence: `4, 7, 10, 13, 16, ...`. A friend says 'the 100th term is 304 because I just did `3 × 100 + 4`.' Is the friend right? If not, give the correct 100th term and explain the fix.

Art hook Plot position on the x-axis and term on the y-axis as dots. The dots fall on a straight line; draw the true line `3x+1` in green and the friend's wrong line `3x+4` in faint red so the constant gap of 3 is visible everywhere.

Continue each add-the-same-amount sequence for two more terms. (a) `10, 20, 30, ...` (b) `1, 4, 7, 10, ...`

Art hook Two number lines stacked; on each, animate a dot hopping by the common difference. Lane (a) hops far, lane (b) hops short, so the fixed step-size shows as a fixed hop length.

For the sequence `6, 11, 16, 21, ...`, what is the common difference (the amount added each time)? Then give the next term.

Art hook Between each pair of dots on a number line, draw a small labelled bracket showing '+5'. All brackets are the same width — a visual proof the difference is constant.

A sequence multiplies by the same amount each step: `1, 3, 9, 27, ...`. What is the rule, and what is the next term?

Art hook Start with 1 dot; each step, replace every dot with 3 dots arranged in a tiny triangle. After 4 steps the screen fills with a self-similar triangular cluster (a Sierpinski-like bloom) — geometric growth made visible.

Run the calculation program 'multiply by 5, then add 1' on the numbers `2` and `6`. What comes out of each?

Art hook An input dot slides into a two-gear machine ('×5' then '+1') and the output dot lands on a number line. Feed several inputs and trace where each output lands.

The table shows a sequence. Copy the pattern of adding the same amount and fill the blank. Position: `1, 2, 3, 4, 5`. Term: `7, 12, 17, __ , 27`.

Art hook Render the table as a row of columns whose heights are the terms; the missing bar is a dashed outline the learner 'fills' by dragging it to height 22, snapping to the straight top edge the other bars form.

For `2, 6, 10, 14, ...` the step is +4 and position 1 gives 2. Someone guesses the position rule is `4 × position`. Test it at position 1: does it give 2? What small change fixes it?

Art hook Plot the true dots and the line `4x`; the true dots sit exactly 2 below the line everywhere. Shade the constant vertical gap of 2 to reveal the '−2' correction.

A sequence has the position rule `3 × position + 2`. (a) Use it to find the 1st, 2nd, and 10th terms directly. (b) Which position gives the term `32`?

Art hook Points (position, term) plotted as dots on a grid; the dots line up straight. Let the learner click any x to reveal the y-dot instantly ('jump to the 10th'), and click a y to trace back down to its position.

Find the rule for `5, 10, 20, 40, ...` and give the next term. (Careful: is it 'add the same amount' or 'multiply by the same amount'?)

Art hook Draw concentric rings whose radii double each step (5, 10, 20, 40 pixels). Doubling makes the gaps grow — a bull's-eye that spreads out faster and faster, contrasting with an even 'add' pattern shown alongside.

An evolving sequence uses 'each new term is the sum of the two terms before it' and starts `1, 3`. Write the next four terms.

Art hook Build growing squares whose side lengths are the terms (1, 3, 4, 7, 11, ...) fitted together in a spiral, and sweep a quarter-circle arc through each — a Fibonacci-style spiral from a look-back rule.

A sequence goes `100, 91, 82, 73, ...`. Describe the rule in words and give the next two terms.

Art hook A vertical number line where a dot falls downward by 9 each frame, leaving a fading trail. Because the step is constant, the trail dots are evenly spaced — a steady descent.

Make your own: choose a start and a 'multiply then add' rule (like 'start at 2, then each term is ×2 + 1'). Write the first four terms and state your rule so a friend could continue it.

Art hook A live 'rule sandbox': sliders for the multiplier and the add-amount, and a colour that shifts as terms grow. Placing each term as a dot on a spiral lets learners see how a steeper rule coils the spiral tighter.

Spot the odd one out and explain: in `2, 4, 6, 8, 11` one term breaks the 'add 2' rule. Which term is wrong, and what should it be?

Art hook Dots placed on a number line at the sequence values; the correct ones glow green, the rule-breaker (11) glows red and is nudged by an arrow to its correct spot (10), snapping into the even spacing.