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Arithmetic progressions Cambridge IGCSE Additional Mathematics revision

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In plain words

5, 8, 11, 14: add 3 each time. A sequence that goes up, or down, by the same amount at every step is an arithmetic progression, and two formulas answer every question about it.

6 things to know

  1. In an arithmetic progression, the difference between one term and the next is constant. It is called the common difference, d. The first term is a.
  2. The nth term is a + (n − 1)d.
  3. The sum of the first n terms is Sₙ = (n/2)(2a + (n − 1)d). If the last term l is known, Sₙ = (n/2)(a + l).
  4. Both formulas are on the formula list.
  5. Given two terms, write an equation for each and subtract to find d.
  6. A sum of terms from the middle of a progression is itself an arithmetic progression: count its terms and use (n/2)(first + last).

Worked example

For the progression 5, 8, 11, …, find the 20th term and the sum of the first 20 terms.

  1. a = 5 and d = 3.
  2. 20th term = 5 + 19 × 3 = 62.
  3. S₂₀ = (20/2)(5 + 62) = 10 × 67 = 670.

Worked example

The 3rd term of an arithmetic progression is 11 and the 8th term is 31. Find the first term and the common difference.

  1. a + 2d = 11 and a + 7d = 31.
  2. Subtract: 5d = 20, so d = 4.
  3. a = 11 − 8 = 3.

Tips and tricks

  • The nth term uses (n − 1), not n. The 20th term is 19 steps on from the first.
  • To show a sequence is arithmetic, show that the difference between consecutive terms is the same each time.
5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Arithmetic progressions: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

  1. What is the common difference of 12, 9, 6, 3, …?
    • 3
    • −3 (the answer)
    • 9
    • 1/3

    Each term is 3 less than the one before.

  2. What is the 10th term of the progression with a = 4 and d = 5?
    • 45
    • 49 (the answer)
    • 50
    • 54

    4 + 9 × 5.

  3. What is the sum of the first 10 terms of 1 + 2 + 3 + …?
    • 45
    • 50
    • 55 (the answer)
    • 100

    (10/2)(1 + 10).

  4. The 2nd term of an arithmetic progression is 7 and the 5th is 16. What is d?
    • 2
    • 3 (the answer)
    • 4
    • 9

    Three steps add 9.

  5. Which sequence is an arithmetic progression?
    • 2, 4, 8, 16
    • 1, 4, 9, 16
    • 3, 7, 11, 15 (the answer)
    • 1, 1, 2, 3

    It has a constant difference of 4.

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