What is a closed form for a sequence?

A formula for un directly in terms of n.

How do you find the auxiliary equation of u{n+1}-5un=8?

Replace the right-hand side by 0 and try un=m^n: m-5=0.

What is the general solution of a first-order recurrence relation?

Complementary function plus a particular solution.

Find the constant particular solution of u{n+1}-5un=8.

\lambda-5\lambda=8, so \lambda=-2.

What are the four stages of a proof by induction?

Basis case, assumption for n=k, inductive step to n=k+1, conclusion.

What is the equilibrium of P{n+1}=1.15Pn-60?

400, from \lambda=1.15\lambda-60.

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

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What is the general form of a first-order recurrence relation in this course?: $u_{n+1}+f(n)u_n=g(n)$
What is a closed form for a sequence?: A formula for $u_n$ directly in terms of $n$ .
How do you find the auxiliary equation of $u_{n+1}-5u_n=8$ ?: Replace the right-hand side by $0$ and try $u_n=m^n$ : $m-5=0$ .
What is the complementary function of $u_{n+1}-5u_n=8$ ?: $A\times5^n$
What is the general solution of a first-order recurrence relation?: Complementary function plus a particular solution.
Which trial form do you use for a particular solution when $g(n)$ is a constant?: $u_n=\lambda$
Which trial form do you use when $g(n)=pn+q$ ?: $u_n=an+b$
Which trial form do you use when $g(n)=kr^n$ and $r$ is not a root of the auxiliary equation?: $u_n=\lambda r^n$
Find the constant particular solution of $u_{n+1}-5u_n=8$ .: $\lambda-5\lambda=8$ , so $\lambda=-2$ .
Solve $u_{n+1}-5u_n=8$ with $u_1=1$ .: $u_n=3\times5^{n-1}-2$
What are the four stages of a proof by induction?: Basis case, assumption for $n=k$ , inductive step to $n=k+1$ , conclusion.
What is the equilibrium of $P_{n+1}=1.15P_n-60$ ?: $400$ , from $\lambda=1.15\lambda-60$ .

Exam questions on First-order recurrence relations

A sequence is defined by $u_{n+1}-5u_n=8$ for $n\geq1$ , with $u_1=1$ .
Hence find $u_n$ in terms of $n$ .2 marks
A fish farm models the number of fish $P_n$ in a lake at the start of year $n$ by $P_{n+1}=1.15P_n-60$ , with $P_1=500$ . Each year the population grows by 15% and then 60 fish are removed.
Given that $P_n=100(1.15)^{n-1}+400$ , find the first year in which the population exceeds $1000$ .2 marks
A sequence satisfies $u_{n+1}-2u_n=6n$ for $n\geq1$ , with $u_1=5$ .
Find the complementary function and a particular solution of the form $u_n=an+b$ .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).