Sequences

Formula for the nth term of a sequence

Example

Calculate the first six terms of the sequence:

\[ u_n = 3n +1 \]

\[ \begin{aligned} \text{The first term } &u_{1} = 3 \times 1 + 1 = 3 + 1 = 4 \\[0.6em] \text{The second term } &u_{2} = 3 \times 2 + 1 = 6 + 1 = 7 \\[0.6em] \text{The third term } &u_{3} = 3 \times 3 + 1 = 9 + 1 = 10 \\[0.6em] \text{The fourth term } &u_{4} = 3 \times 4 + 1 = 12 + 1 = 13 \\[0.6em] \text{The fifth term } &u_{5} = 3 \times 5 + 1 = 15 + 1 = 16 \\[0.6em] \text{The sixth term } &u_{6} = 3 \times 6 + 1 = 18 + 1 = 19 \end{aligned} \]

\[ \begin{align*} u_1 &= 4 \\ u_2 &= 7 \\ u_3 &= 10 \\ u_4 &= 13 \\ u_5 &= 16 \\ u_6 &= 19 \end{align*} \]

For what value of \(n\) is: \( u_n = 46 \)

\[ \begin{aligned} 3n + 1 &= 46 \\ 3n &= 45 \\ n &= 15 \end{aligned} \]

Recurrence Relationships

A recurrence relation describes a sequence in which each term is a function of a previous term.

Example

\[ \begin{aligned} \text{Given } &u_{n+1} = u_n + 6 \quad\text{starting with } u_1 = 3 \\[1em] \text{The first term is } &u_1 = 3 \\[0.6em] \text{The second term is } &u_2 = u_1 + 6 = 3 + 6 = 9 \\[0.6em] \text{The third term is } &u_3 = u_2 + 6 = 9 + 6 = 15 \\[0.6em] \text{The fourth term is } &u_4 = u_3 + 6 = 15 + 6 = 21 \\[0.6em] \text{The fifth term is } &u_5 = u_4 + 6 = 21 + 6 = 27 \\[0.6em] \text{The sixth term is } &u_6 = u_5 + 6 = 27 + 6 = 33 \\[1em] \text{Giving the sequence } &3,\,9,\,15,\,21,\,27,\,33 \end{aligned} \]

Example

\[ u_1 = 4,\qquad u_{n+1} = u_n + 3 \] \[ 4,\; 7,\; 10,\; 13,\; 16,\; \dots \]

Example

\[ u_1 = 2,\qquad u_{n+1} = 2u_n \] \[ 2,\; 4,\; 8,\; 16,\; 32,\; \dots \]

Example

\[ \begin{aligned} \text{Write down a recurrence relation for the sequence } &20,\,17,\,14,\,11 \\[1em] u_1 &= 20 \\[0.6em] u_{n+1} &= u_n - 3 \end{aligned} \]

Example

An art dealer expects a painting worth £1 million to appreciate by 5% per year.

Find a recurrence relation for the value of the painting.
Calculate the value after 5 years.

\[ \begin{aligned} u_{n+1} &= 1.05\,u_n \quad\text{starting with } u_0 = \text{£}1\text{ million} \\[1em] \text{The value at the end of year 1 is } &u_1 = 1.05 \times 1 = 1.05 \\[0.6em] \text{The value at the end of year 2 is } &u_2 = 1.05 \times 1.05 = 1.1025 \\[0.6em] \text{The value at the end of year 3 is } &u_3 = 1.05 \times 1.1025 = 1.157625 \\[0.6em] \text{The value at the end of year 4 is } &u_4 = 1.05 \times 1.157625 = 1.21550625 \\[0.6em] \text{The value at the end of year 5 is } &u_5 = 1.05 \times 1.21550625 = 1.276281563 \\[1em] \text{After 5 years the painting is worth } &\text{£}1.28\text{ million} \end{aligned} \]

Example

£50,000 is borrowed at 6.5% p.a. with monthly repayments of £322.50.

How much is owed after 5 years?
How long does it take to pay off the loan?

\[ \begin{aligned} \text{Monthly repayments are } &\text{£}322.50 = \text{£}3870 \text{ per annum} \\[1em] u_{n+1} &= 1.065\,u_n - 3870 \quad\text{starting with } u_0 = \text{£}50{,}000 \\[1em] \text{End of year 1: } &u_1 = 1.065 \times 50000 - 3870 = 49380 \\[0.6em] \text{End of year 2: } &u_2 = 1.065 \times 49380 - 3870 = 48719.70 \\[0.6em] \text{End of year 3: } &u_3 = 1.065 \times 48719.70 - 3870 = 48016.4805 \\[0.6em] \text{End of year 4: } &u_4 = 1.065 \times 48016.4805 - 3870 = 47267.55173 \\[0.6em] \text{End of year 5: } &u_5 = 1.065 \times 47267.55173 - 3870 = 46469.9426 \\[1em] \text{After 5 years, } &\text{£}46{,}469.94 \text{ is owed.} \end{aligned} \]

It takes 30 years to repay the loan.

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