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Linear second order recurrence sequence

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Find a closed form for

uo = 2
u1 = 7
un + 2 = 2un + 1 + 8un (n = 0,1,2,3,.......)
(The o,1 and n terms are subscript).
The techniques given is using a auxillary equation - finding a generalised solution where A and B are unknown constants. Finding A and B using the initial terms:

Auxillary equation
r^2 - 2r - 8
(r - 4)(r + 2)
r = 4 and r = -2

The general solution
un = A4^n + B(-2)^n where A and B are unknown constants
To find A + B use the initial terms
uo = 2 gives A + B = 2
u1 = 7 gives 4A - 2B =7

Hence A = 11/6 and B = 1/6 so

un = 11/6(4)^n + 1/6(-2)^n (n = 0,1,2.....)

Thank you

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Solution Summary

The expert examines linear second order recurrence sequences.

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You have got two equations

A + B = 2------------(i)
4A - 2B =7----------(ii)

Multiply (i) by 2 on each side, we will ...

Solution provided by:
Sandhya Sharma, MSc
Education
  • BSc, Meerut University
  • MSc, Meerut University
  • MPhil, Institute of Advanced Studies
  • MSc, AIT
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