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Recurrence relations

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Sequences based on recurrence relations

In maths, a sequence is an ordered set of numbers. For example 1, 5, 9, 13, 17. For this sequence, the rule is add four.

Each number in a sequence is called a term and is identified by its position within the sequence. We write them as follows.

  • The first term U_1 = 1

  • The second term U_2 = 5

  • The third term U_3 = 9

  • The nth term U_n

The above sequence can be generated in two ways.

  1. You can use a formula for the nth term. Here it would be U_n = 4n - 3. Adding the same amount (in this case 4) generates each term. Each term will therefore be a multiple of 4 \Rightarrow 4n.

    However, the first term when n=1 is 1.

    4(1) + ? = 1

    4(1) - 3 = 1

    When n=1U_1 = 4(1) - 3 = 1

    When n =2U_2 = 4(2) - 3 = 5 and so on.

  2. The other way of generating this sequence is by using a recurrence relation, where each term is generated from the previous value.

    When n=1U_1 = 1

    When n=2U_2 = 1 + 4 = 5

    When n=3U_2 = 5 + 4 = 9

    The recurrence relation would therefore be U_{n + 1} = U_n + 4. The starting value, U_1, would have to be provided. Note that the starting value can also be {\rm{U}}_{\rm{0}} .

A recurrence relation is a sequence that gives you a connection between two consecutive terms. These two terms are usually U_{n + 1} and U_n. However they could be given as U_n and U_{n - 1}

Generally, the linear recurrence relation U_{n + 1} = a\;U_n + b

Question

A sequence is given by the recurrence relation U_{n + 1} = 3\;U_n + 9.

If U_0 = - 4 find the first 5 terms

Given U_{n + 1} = 3\;U_n + 9 you can work out that

toggle answer

Answer

Step 1:

When n = 0 U0 = -4

U_{0 + 1} = 3\;U_0 + 9 \Rightarrow U_1 = 3( - 4) + 9 = - 3 \Rightarrow U_1 = - 3

Step 2:

When n = 1 U1 = -3

U_{1 + 1} = 3\;U_1 + 9 \Rightarrow U_2 = 3( - 3) + 9 = 0 \Rightarrow U_2 = 0

Step 3:

When n = 2 U2 = 0

U_{2 + 1} = 3\;U_2 + 9 \Rightarrow U_3 = 3(0) + 9 = 0 \Rightarrow U_3 = 9

Step 4:

When n = 03 U3 = 9

U_{3 + 1} = 3\;U_3 + 9 \Rightarrow U_4 = 3(9) + 9 = 36 \Rightarrow U_4 = 36

So the first 5 terms are -4, -3, 0, 9, 36

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