How do you find the particular solution for a constant right-hand side?

Try un=\lambda, so \lambda-a\lambda=b and \lambda=\dfrac{b}{1-a}.

What is the general solution of a non-homogeneous relation?

Complementary function plus particular solution.

Substitute the initial condition (e.g. u1) into the general solution.

Solve u{n+1}-5un=8, u1=1.

un=\frac35\times5^n-2=3\times5^{n-1}-2.

What does un approach if |a|<1?

The particular solution \lambda=\dfrac{b}{1-a}, because a^n\to0.

What happens if |a|1?

The complementary function grows, so un diverges.

What must you do when dividing an inequality by \ln0.75?

Reverse the inequality, because \ln0.75 is negative.

How can you check a solution quickly?

Compute u2 from the relation and from the formula; they must agree.

First-order recurrence relationsEdexcel A-Level Further Maths: Flashcards

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What is a first-order linear recurrence relation?

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What is a first-order linear recurrence relation?: A rule giving $u_{n+1}$ from $u_n$ alone, with $u_n$ to the first power, e.g. $u_{n+1}=au_n+b$ .
What is the auxiliary equation for $u_{n+1}-a\,u_n=0$ ?: $m-a=0$ , giving $m=a$ .
What is the complementary function for $u_{n+1}-a\,u_n=b$ ?: $A\,a^n$ , where $A$ is a constant.
How do you find the particular solution for a constant right-hand side?: Try $u_n=\lambda$ , so $\lambda-a\lambda=b$ and $\lambda=\dfrac{b}{1-a}$ .
What is the general solution of a non-homogeneous relation?: Complementary function plus particular solution.
How do you find $A$ ?: Substitute the initial condition (e.g. $u_1$ ) into the general solution.
Solve $u_{n+1}-5u_n=8$ , $u_1=1$ .: $u_n=\frac35\times5^n-2=3\times5^{n-1}-2$ .
What does $u_n$ approach if $|a|<1$ ?: The particular solution $\lambda=\dfrac{b}{1-a}$ , because $a^n\to0$ .
What happens if $|a|>1$ ?: The complementary function grows, so $u_n$ diverges.
How do you write 'grows by 10% then 30 are added'?: $u_{n+1}=1.1u_n+30$ .
How do you write 'loses 25% then a dose of $d$ is added'?: $u_{n+1}=0.75u_n+d$ .
What must you do when dividing an inequality by $\ln0.75$ ?: Reverse the inequality, because $\ln0.75$ is negative.
How can you check a solution quickly?: Compute $u_2$ from the relation and from the formula; they must agree.

Exam questions on First-order recurrence relations

A pond is stocked with $400$ fish at the start of year $1$ . Each year the number of fish increases by $10\%$ and then $30$ more fish are added. Let $u_n$ be the number of fish at the start of year $n$ , so that $u_{n+1}=1.1u_n+30$ with $u_1=400$ .
Find the particular solution of the form $u_n=\lambda$ .2 marks
A loan of $\pounds5000$ is repaid monthly. Each month $0.5\%$ interest is added to the amount owed and then a repayment of $\pounds200$ is made. Let $u_n$ be the amount owed, in pounds, after $n$ repayments, so $u_0=5000$ .
Hence solve the recurrence relation to find $u_n$ in terms of $n$ .2 marks
The number of bacteria in a culture, in thousands, at the start of day $n$ is $u_n$ . Each day the number quadruples and then $9$ thousand bacteria are removed for testing, so $u_{n+1}=4u_n-9$ with $u_1=5$ .
Solve the recurrence relation to find $u_n$ in terms of $n$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).