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

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

What must an induction proof for a second-order relation include?

Base cases n=1 and n=2; assume n=k and n=k+1; prove n=k+2.

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

Card 1 of 120 of 12 known

Question

What is the general form of a second-order recurrence relation?

Tap or press Space to reveal

Tap card or press Space to flip

See all 12 cards

What is the general form of a second-order recurrence relation?: $u_{n+2}+f(n)u_{n+1}+g(n)u_n=h(n)$
How many initial conditions do you need for a second-order relation?: Two, e.g. $u_1$ and $u_2$ .
What is the auxiliary equation for $au_{n+2}+bu_{n+1}+cu_n=0$ ?: $am^2+bm+c=0$
What is the complementary function for two distinct real roots $m_1$ , $m_2$ ?: $A\,m_1^n+B\,m_2^n$
What is the complementary function for a repeated root $m$ ?: $(A+Bn)m^n$
Find the roots for $u_{n+2}-7u_{n+1}+12u_n=0$ .: $m^2-7m+12=0$ , so $m=3$ or $m=4$ .
Which trial particular solution is used when $h(n)$ is a constant?: $u_n=\lambda$
Which trial particular solution is used when $h(n)$ is linear?: $u_n=an+b$
What do you do if the trial particular solution already appears in the complementary function?: Multiply the trial form by $n$ .
Find the constant particular solution of $2u_{n+2}+7u_{n+1}-15u_n=6$ .: $-6\lambda=6$ , so $\lambda=-1$ .
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 must an induction proof for a second-order relation include?: Base cases $n=1$ and $n=2$ ; assume $n=k$ and $n=k+1$ ; prove $n=k+2$ .

Exam questions on Second-order recurrence relations

A sequence satisfies $u_{n+2}-7u_{n+1}+12u_n=0$ for $n\geq1$ , with $u_1=2$ and $u_2=2$ .
Find $u_n$ in terms of $n$ .2 marks
A sequence satisfies $u_{n+2}-6u_{n+1}+9u_n=16$ for $n\geq1$ , with $u_1=10$ and $u_2=31$ .
Find $u_n$ in terms of $n$ .2 marks
A sequence satisfies $u_{n+2}-5u_{n+1}+6u_n=2n+1$ for $n\geq1$ , with $u_1=8$ and $u_2=17$ .
Find a particular solution of the form $u_n=an+b$ .3 marks

See the full worksheet

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).