Quick questions on Recurrence relations explained: H2 Further Mathematics

6short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What are first-order linear recurrences?

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A first-order linear recurrence has the form

u_{n+1} = a\,u_n + b

with constants

a

and

b

What are non-homogeneous second-order recurrences?

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If the recurrence has an extra term,

u_{n+2} = p\,u_{n+1} + q\,u_n + f(n)

, add a particular solution matching the form of

f(n)

(a constant for constant

f

, a linear

cn + d

for linear

f

, and so on) to the homogeneous solution, then apply the initial conditions last.

What is index confusion?

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Be careful whether the sequence starts at

u_0

u_1

; the initial conditions must match the stated starting index.

What is q1?

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Write the characteristic equation of

u_{n+2} = u_{n+1} + 6u_n

and find its roots. [2 marks]

What is q2?

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Give the general solution of a second-order recurrence whose characteristic equation has a repeated root

\lambda = 5

. [1 mark]

What is q3?

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For

u_{n+1} = 2u_n + 6

, find the constant particular solution. [2 marks]

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