Suppose that the sequence x0, x1, x2... is defined by x0 = 1, x1 = 4, and xk+2 = −6xk+1−8xk for k≥0. Find a general formula for xk. Be sure to include parentheses where necessary, e.g. to distinguish 1/(2k) from 1/2k. .
No idea how to go about this, please help
If I may ask a follow up question:
everything makes perfect sense up until the last line. How exactly did you come up with -3 and 4? I've been trying to solve similar problems and the coefficient seems the only thing off
x0 = 1
x1 = 4
x2 = -6*4 - 8*1 = -32
x3 = -6(-32)-8*4 = 160
x4 = -6(160)-8(-32) = -704
x5 = -6(-704)-8(160) = 2944
Read about characteristic functions for recursive sequences. For example, at
www.csee.umbc.edu/~stephens/203/PDF/8-3.pdf
It shows that the characteristic equation for this sequence is
x^2 = -6x-8
x^2+6x+8 = 0
(x+4)(x+2) = 0
x = -4,-2
So, the general formula for the sequence is
xn = C(-4)^n+D(-2)^n
plugging x0 and x1, we get
xn = -3(-4)^n + 4(-2)^n
You can see that it works for the terms given above.
This article is also useful.
https://math.stackexchange.com/questions/731916/strategies-for-developing-explicit-formulas-for-nth-term-given-recurrence-relati
remember: google is your friend!
I am so lost, how did you get -3 and +4? Also, the PDF is not opening for me, so I'm unable to see what you have shared.
nevermind, the pdf shows it very clearly
Well, let's break it down step by step and see if we can find a pattern. We're given that x0 = 1 and x1 = 4.
Now, if we apply the recursion formula xk+2 = −6xk+1 − 8xk and substitute k = 0, we get:
x2 = −6x1 − 8x0
x2 = −6(4) − 8(1)
x2 = -24 - 8
x2 = -32
Similarly, if we substitute k = 1, we get:
x3 = −6x2 − 8x1
x3 = −6(-32) − 8(4)
x3 = 192 - 32
x3 = 160
Continuing this pattern, we can see that:
x4 = −6x3 − 8x2
x5 = −6x4 − 8x3
x6 = −6x5 − 8x4
...
So let's try to generalize this. It seems like each term is a combination of the two previous terms.
Let's define a new sequence, yk, where yk = xk+1. So now we have:
y0 = x1 = 4
y1 = x2 = -32
y2 = x3 = 160
...
Now, notice that each term yk can be obtained using the formula:
yk = −6yk-1 − 8yk-2
where k ≥ 1.
So, we can write a general formula for yk. Using the notation yk+2 = −6yk+1 − 8yk, we have:
yk = yk-2 = −6yk-3 − 8yk-4
...
y2 = y0 = −6y-1 − 8y-2
Now, substituting yk = xk+1, we get:
xk+1 = xk-1 = −6xk-2 − 8xk-3
...
x3 = x1 = −6x0 − 8x-1
And since we know that x0 = 1, we can substitute it in:
x3 = x1 = −6(1) − 8x-1
So, the general formula for the sequence xk is:
xk = −6xk-2 − 8xk-3, for k ≥ 1
Hope that helps! If not, at least we had a fun little logic exercise!
To find a general formula for the sequence x0, x1, x2..., we can use the method of generating functions.
First, let's define the generating function f(x) for the sequence x0, x1, x2... as:
f(x) = x^0 * x0 + x^1 * x1 + x^2 * x2 + ...
In this case, we have x0 = 1, x1 = 4, and the recursion relation xk+2 = -6xk+1 - 8xk for k≥0. This allows us to write the generating function equation as follows:
f(x) = 1 + 4x + (-6x)(f(x)) + (-8x^2)(f(x))
Let's simplify this equation step by step:
f(x) = 1 + 4x - 6xf(x) - 8x^2f(x)
Rearranging the equation, we get:
f(x) + 6xf(x) + 8x^2f(x) = 1 + 4x
Now, let's factor f(x) out:
f(x)(1 + 6x + 8x^2) = 1 + 4x
Divide both sides of the equation by (1 + 6x + 8x^2):
f(x) = (1 + 4x) / (1 + 6x + 8x^2)
Now, let's express (1 + 6x + 8x^2) as a product of two factors:
f(x) = (1 + 4x) / [(1 + 4x)(1 + 2x)]
Notice that we can cancel out (1 + 4x) terms on the numerator and denominator:
f(x) = 1 / (1 + 2x)
Now, we will expand the right-hand side into a power series using the formula for a geometric series:
f(x) = 1 / (1 + 2x) = 1 - 2x + 4x^2 - 8x^3 + 16x^4 - ...
Therefore, the general formula for the sequence x0, x1, x2... is:
xk = [coefficient of x^k in f(x)] = coefficient of x^k in (1 - 2x + 4x^2 - 8x^3 + 16x^4 - ...)
From the expression above, we can see that the coefficient of x^k is (-2)^k.
So the general formula for xk is:
xk = (-2)^k
Therefore, the general formula for the sequence x0, x1, x2... is xk = (-2)^k.