Maths Olympiad · topic 29 of 33 · free

Sequences and AP — Maths Olympiad practice

In an arithmetic progression (AP) the numbers go up by the same step every time. To reach the nth term you take n − 1 steps, not n — that one slip is behind most wrong options. Sums are quick if you pair the first and last terms.

How to solve it

Same step every time — “an AP grows by the same amount, so count the steps”

  1. In an AP the gap (common difference d) is always the same.
  2. nth term = a + (n − 1) × d, where a is the first term.
  3. Sum of n terms = n × (first + last) ÷ 2; so 1 + 2 + … + n = n × (n + 1) ÷ 2, and the first n odd numbers add up to n × n.

Use it for: Number patterns, savings plans, seating rows and AP questions in Class 8–10 Olympiad papers.

Worked example: What is the 20th term of the AP 3, 7, 11, 15, …? → 79

  1. First term 3, common difference 4.
  2. 20th term = 3 + 19 × 4 = 3 + 76 = 79.

Check: 79 − 3 = 76 = 19 steps of 4, and the 1st term plus 19 steps is the 20th term ✓

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