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I have the answer to the following question.

Solve the following recurrence equation with the given initial values:
$b_n = 4b_{n−1} − 4b_{n−2}$. Initial values: $b_0 = 3$, $b_1 = 10$.

I am looking for an explanation. The answer is: $3\cdot2^n + n2^{n+1}$. How did it get to this answer? Here is where I am at:

$b_n-4b_{n-1}+4b_{n-2}=0$.
$x^2-4x+4=0\implies x=2$.

amWhy
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MBHai2021
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    Welcome to Math.SE! ... The community prefers/expects a question to include something of what the asker knows about the problem. (What have you tried? Where did you get stuck? etc) This helps answerers tailor their responses to best serve you, without wasting time (theirs or yours) explaining things you already understand or using techniques beyond your skill level. (It also helps convince people that you aren't simply trying to get them to do your homework for you. An isolated problem statement with no evidence of personal effort makes a poor impression, attracting down- and close-votes.) – Blue Dec 18 '21 at 19:35
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    Do you know how to solve recurrence relations using generating functions? If so, can you show us how far you got? – Bonnaduck Dec 18 '21 at 19:36
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    See https://en.wikipedia.org/wiki/Recurrence_relation#Solving_homogeneous_linear_recurrence_relations_with_constant_coefficients – lhf Dec 18 '21 at 19:53
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    If you're unfamiliar with how we handle repeated roots, determine the recursion relation for $c_n:=b_n/2^n$. What do you get? – J.G. Dec 18 '21 at 21:28
  • See https://math.stackexchange.com/q/4205834/399263 – zwim Dec 18 '21 at 22:52
  • Here is a method for solving $all$ such problems: https://math.stackexchange.com/questions/2299076/what-is-the-name-of-the-unique-numbers-in-fibonacci-like-integer-sequences/2300458#2300458. – Cye Waldman Dec 19 '21 at 16:14
  • my problem was with repeated roots, so my question is answered now – MBHai2021 Dec 19 '21 at 18:24
  • I added an old Question as duplicate target which clearly stated the repeated root of characteristic equation issue. – hardmath Dec 20 '21 at 01:00

1 Answers1

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$b_n = 4b_{n−1} − 4b_{n−2}$ and then $b_n-4b_{n-1}+4b_{n-2}=0$

Chracteristic equation is $r^2-4r+4=0$ $\implies$ $(r-2)^2=0$ $\quad$

$r_1=2$$\quad$and $r_2=2$$\implies$Since root has repeated, the form is, $b_n=a2^n+bn2^n$

Given $b_0 = 3$ $\implies$ $a+0=3$ and then $a=3$

$b_1 = 10$$\implies$ $10 = 2a + 2b$ $\implies$ $a+b=5$

Substituting the value $a = 3$ $\implies$ $3 + b = 5$ $\implies$ $b=2$

The solution is $b_n = 3(2^n) + (2n)2^n$

J.G.
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