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IGCSE Maths · Edexcel 4XMA1 · Unit 2 Higher · Sequences

Arithmetic Sequences and Series

An arithmetic sequence goes up or down by the same amount each time. Edexcel Higher goes beyond the nth term and asks you to add up the terms, which is called an arithmetic series. One formula, used carefully, covers almost every question.

1Arithmetic sequences: a and d

In an arithmetic sequence the gap between consecutive terms is constant. We use two letters:

  • a = the first term
  • d = the common difference (term − previous term)
Sequencead
7, 11, 15, 19, …74
20, 17, 14, 11, …20−3
2.5, 3, 3.5, 4, …2.50.5

2The nth term

nth term of an arithmetic sequenceun = a + (n − 1)d

To reach the nth term you start at a and add d a total of (n − 1) times.

Worked example
Find the nth term and the 50th term of 7, 11, 15, 19, …
  1. a = 7 and d = 4.
  2. un = 7 + (n − 1) × 4 = 4n + 3.
  3. u50 = 4(50) + 3 = 203.
un = 4n + 3; u50 = 203
Worked example
Is 150 a term of the sequence 4, 11, 18, 25, …?
  1. a = 4, d = 7, so un = 7n − 3.
  2. Set 7n − 3 = 150 → 7n = 153 → n = 21.86…
  3. n must be a whole number, so 150 is not a term.
No: n is not an integer

3The sum of an arithmetic series

The story goes that young Carl Gauss added 1 + 2 + … + 100 in seconds. He wrote the sum forwards and backwards and paired the terms: each pair adds to 101, and there are 100 pairs, which counts every term twice. So the sum is 100 × 101 ÷ 2 = 5050.

Sum of the first n termsSn = n2[2a + (n − 1)d]    or    Sn = n2(a + l), where l is the last term
Worked example
Find the sum of the first 20 terms of 3 + 7 + 11 + …
  1. a = 3, d = 4, n = 20.
  2. S20 = 202[2(3) + 19(4)] = 10 × (6 + 76) = 10 × 82.
S20 = 820
Worked example
Find the sum 6 + 10 + 14 + … + 86.
  1. First find how many terms: 6 + (n − 1)4 = 86 → 4(n − 1) = 80 → n = 21.
  2. Use the a + l form: S21 = 212(6 + 86) = 212 × 92 = 966.
966
Exam tipWhen you know the last term, find n first using un = a + (n − 1)d. A wrong number of terms is the most common error.

4Using two terms to find a and d

Worked example
In an arithmetic sequence the 5th term is 17 and the 12th term is 38. Find the sum of the first 20 terms.
  1. u5: a + 4d = 17. u12: a + 11d = 38.
  2. Subtract: 7d = 21 → d = 3. Then a = 17 − 12 = 5.
  3. S20 = 202[2(5) + 19(3)] = 10 × 67 = 670.
670

5Finding n from a sum

If you are told the sum and asked for n, the formula gives a quadratic in n. Solve it and keep the positive whole-number solution.

Worked example
How many terms of 5 + 8 + 11 + … are needed to make a total of 440?
  1. a = 5, d = 3: Sn = n2[10 + 3(n − 1)] = n2(3n + 7).
  2. Set equal to 440: n(3n + 7) = 880 → 3n² + 7n − 880 = 0.
  3. Factorise: (n − 16)(3n + 55) = 0, so n = 16 or n = −55/3.
  4. n must be a positive integer.
16 terms

6Arithmetic series in real life

Anything that increases by a fixed amount each step is an arithmetic sequence: savings that grow by the same amount each month, seats in rows of a theatre, or stacked cans in a display.

Worked example
Maya saves $500 in the first month and $150 more each month than the month before. How much has she saved in total after 12 months?
  1. a = 500, d = 150, n = 12.
  2. S12 = 6[2(500) + 11(150)] = 6 × 2650.
$15 900
Common mistakes
  • Using n instead of (n − 1) in un = a + (n − 1)d.
  • Getting the number of terms wrong when you are given the last term. Find n first.
  • Mixing up un (one term) and Sn (the total of n terms).
  • Keeping a negative or non-integer n when solving for the number of terms.
  • Forgetting the brackets: Sn = n/2 × [2a + (n − 1)d], not n/2 × 2a + (n − 1)d.

★Key Vocabulary Recap

Arithmetic sequence
A sequence with a constant difference between terms.
Common difference d
The amount added each time (can be negative).
Series
The sum of the terms of a sequence.
un
The nth term.
Sn
The sum of the first n terms.
Specification reference (Pearson Edexcel 4XMA1, Unit 2 Higher)
  • Use the common difference and first term of an arithmetic sequence.
  • Find the nth term of an arithmetic sequence.
  • Find the sum of the first n terms of an arithmetic series (Sn).
Next stepReady to practise? Try the 12 exam-style questions, then check your working against the worked solutions.
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