Sequences and recurrence relations

nth term formula

u_n = f(n) gives any term directly. For u_n = n² - 3n, the 5th term is 25 - 15 = 10.

Recurrence relations

u_n+1 = f(u_n) gives each term from the previous one, so you also need a starting value such as u_1 = 2. Work through the terms one at a time.

Types of sequence

• Increasing: u_n+1 > u_n for all n
• Decreasing: u_n+1 < u_n for all n
• Periodic: the terms repeat in a cycle. The order (period) is the number of terms in the cycle.

Worked example

u_1 = 2 and u_n+1 = 3u_n - 2. Find u_2, u_3 and u_4.

1. u_2 = 3(2) - 2 = 4
2. u_3 = 3(4) - 2 = 10
3. u_4 = 3(10) - 2 = 28

Answer: 4, 10, 28

Check yourself

1. u_n = n² - 3n. Find u_5.
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   10. 25 - 15 = 10.

2. u_1 = 1 and u_n+1 = 2u_n + 3. Find u_4.
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   29. u_2 = 5, u_3 = 13, u_4 = 29.

3. u_1 = 2 and u_n+1 = 1/(1 - u_n). Find the order (period) of this sequence.
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   3. u_2 = 1/(1 - 2) = -1, u_3 = 1/(1 + 1) = 1/2, u_4 = 1/(1 - 1/2) = 2. The terms repeat every 3.

4. For the same sequence (u_1 = 2, u_n+1 = 1/(1 - u_n)), find u_100.
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   2. The cycle 2, -1, 1/2 has order 3. 100 = 3 x 33 + 1, so u_100 = u_1 = 2.

5. Which sequence is increasing?
   • u_n = 3n - 7
   • u_n = 10 - n
   • u_n = (-1)ⁿ n
   • u_n = 1/n
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   u_n = 3n - 7. Each term is 3 more than the one before.

6. u_1 = 2, u_2 = 10 and u_n+1 = ku_n + 4. Find k.
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   3. u_2 = 2k + 4 = 10, so k = 3.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 2 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.