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

Substitute un=\lambda into all terms and solve for \lambda.

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

Complementary function plus particular solution.

How do you find A and B?

Put the two initial conditions into the general solution and solve simultaneously.

Solve 2u{n+2}+7u{n+1}-15un=6 with u1=10, u2=-17.

un=4\left(\frac32\right)^n-(-5)^n-1.

What does un approach if all roots satisfy |m|<1?

The particular solution \lambda.

How can you check your formula?

Find u3 from the recurrence and from the formula; they must agree.

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

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What is a second-order linear recurrence relation?

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What is a second-order linear recurrence relation?: One linking $u_{n+2}$ , $u_{n+1}$ and $u_n$ with constant coefficients.
How many initial conditions are needed?: Two, for example $u_1$ and $u_2$ .
What is the auxiliary equation of $p\,u_{n+2}+q\,u_{n+1}+r\,u_n=0$ ?: $p\,m^2+q\,m+r=0$ .
Complementary function for distinct real roots $m_1,m_2$ ?: $A\,m_1^n+B\,m_2^n$ .
Complementary function for a repeated root $m$ ?: $(A+Bn)\,m^n$ .
How do you find the particular solution for a constant right-hand side?: Substitute $u_n=\lambda$ into all terms and solve for $\lambda$ .
What is the general solution of the non-homogeneous relation?: Complementary function plus particular solution.
How do you find $A$ and $B$ ?: Put the two initial conditions into the general solution and solve simultaneously.
Roots of $2m^2+7m-15=0$ ?: $m=\frac32$ and $m=-5$ .
Solve $2u_{n+2}+7u_{n+1}-15u_n=6$ with $u_1=10$ , $u_2=-17$ .: $u_n=4\left(\frac32\right)^n-(-5)^n-1$ .
What does $u_n$ approach if all roots satisfy $|m|<1$ ?: The particular solution $\lambda$ .
How can you check your formula?: Find $u_3$ from the recurrence and from the formula; they must agree.

Exam questions on Second-order recurrence relations

A sequence satisfies $u_{n+2}=5u_{n+1}-6u_n$ with $u_1=5$ and $u_2=13$ .
Find $u_n$ in terms of $n$ .2 marks
A sequence satisfies $u_{n+2}=6u_{n+1}-9u_n$ with $u_1=3$ and $u_2=18$ .
Find $u_n$ in terms of $n$ .2 marks
A sequence satisfies $u_{n+2}-u_{n+1}-6u_n=12$ with $u_1=2$ and $u_2=20$ .
Find the complementary function and a particular solution of the form $u_n=\lambda$ .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).