Math · Stage 5

Find and use the nth-term rule of a sequence; sum an arithmetic run

Two Olympiad tools for number sequences: a rule that jumps straight to any term without listing them all, and Gauss's pairing trick that adds a whole run in one line.

✦ Math📘 Stage 5 · S5⏱ ~12 min🧒 Ages 9–10★ Free to start

I can find and use the nth-term rule of a sequence; sum an arithmetic run.

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Kavi and the Staircase of Numbers 🪜

Kavi climbs a staircase where every step rises the same amount 🪜. "If each step adds the same jump," she hoots, "I don't have to climb all the way — I can leap straight to the 100th step with one rule. And when I want the whole staircase at once, I pair the bottom step with the top, the next with the next-to-top… each pair is equal, so I add them all in a single breath!"

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What your child will learn

1Every arithmetic sequence has one steady stepHere is the run 3, 7, 11, 15, 19, 23. The lit numbers are the terms; the dim ones are skipped. From each term to the next we add the same amount. That steady amount is the common difference — the first thing to find.

2The nth-term rule: jump straight to any positionGive every term a position n. The 1st term is the start; each step after adds the common difference. So the nth term = first + (n − 1) × step. Tap a row to hear it.

3Write the rule as T(n) = step × n + shiftOnce you know the step, the rule has the shape T(n) = step × n + shift. Pick the shift so the 1st term comes out right, then you can leap to any n. Tap each card.

4Add a whole run by pairing the endsTo add 1 + 2 + 3 + … + 10 the slow way is nine additions. Gauss paired the ends instead: first with last, second with second-last… each pair is equal. Tap each card.

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A peek inside

A sequence starts 2, 5, 8, 11, … (step 3). What is the 10th term?

nth term = first + (n − 1) × step = 2 + (10 − 1) × 3 = 2 + 27

Try this at home: Next term: 6, 11, 16, 21, __ (common difference 5)

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Start the free interactive lesson, or print the handout for screen-free play.

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Part of Math · Stage 5 · S5 on KAVI Academy — a secular, whole-child learning platform for ages 5–12.