Find an expression for a sequence involving the sum of nth powers

In summary, the problem was that the equations were not typed properly and the solution was found by converting a series into one equation.
  • #1
rugerts
153
11
Homework Statement
a_n = 2*a_n-1 + a_n-2 -2*a_n-3.
a_0 = 5
a_1 = 18
a_2 = 14
Relevant Equations
No real equations. Shown below is a similar example done in class that I'm trying to base my solution around.
Example done in class:
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The problem and my solution:
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My solution seems incorrect because if I try to plug in 0, I don't get the initial condition given in the problem.

Does anyone see what I've done wrong along the way?

Thanks.
 

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  • #2
The equations would be much easier to read when typed here, with explanations what you do why.
$$a_n = \frac{5^n}{5} + \frac{7^n}{7} + \frac 2 3 3^n$$
You shouldn't get three different equations for an+1, an+2 and an+3.

Where do you plug in your initial conditions? You should have a general solution with three unknowns (which looks very different from what you got), these unknowns can be computed based on the initial conditions.
 
  • #3
mfb said:
The equations would be much easier to read when typed here, with explanations what you do why.
$$a_n = \frac{5^n}{5} + \frac{7^n}{7} + \frac 2 3 3^n$$
You shouldn't get three different equations for an+1, an+2 and an+3.

Where do you plug in your initial conditions? You should have a general solution with three unknowns (which looks very different from what you got), these unknowns can be computed based on the initial conditions.
Did you look through the first example I was shown in lecture? That general solution only has powers of n in it, which I think is the goal. Maybe I'm misunderstanding you.
 
  • #4
Not been through all your calculation, but I think you can do a quite simpler calculation in this case if you define a new variable, either
bn = an - 2an-1
or probably better
cn = an - an-2.

You then get a very simple series for the new variable, which you can convert into one, well maybe you should call it two, for the an .
 
  • #5
epenguin said:
Not been through all your calculation, but I think you can do a quite simpler calculation in this case if you define a new variable, either
bn = an - 2an-1
or probably better
cn = an - an-2.

You then get a very simple series for the new variable, which you can convert into one, well maybe you should call it two, for the an .
Interesting. I hate to turn down simpler solutions but I think, since this is a course in linear algebra and we've just covered diagonalization, that we're expected to solve this using that technique.
 
  • #6
rugerts said:
Did you look through the first example I was shown in lecture? That general solution only has powers of n in it, which I think is the goal. Maybe I'm misunderstanding you.
If you consider powers of 0 and -1 it will work...
 
  • #7
So I actually had a mistake in matrix multiplication. I've got the correct solution now. My approach was fine. Thanks all.
 
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Related to Find an expression for a sequence involving the sum of nth powers

1. What does it mean to find an expression for a sequence involving the sum of nth powers?

Finding an expression for a sequence involving the sum of nth powers means to write a formula or equation that can be used to calculate the sum of the first n powers of a given sequence of numbers.

2. How is the sum of nth powers useful in mathematics and science?

The sum of nth powers has many applications in mathematics and science, particularly in areas such as statistics, calculus, and physics. It can be used to calculate areas under curves, solve differential equations, and model real-world phenomena.

3. Can you provide an example of a sequence involving the sum of nth powers?

One example of a sequence involving the sum of nth powers is the Fibonacci sequence, where each term is the sum of the two previous terms. The expression for this sequence involving the sum of nth powers is (1^n + 2^n + 3^n + ... + n^n).

4. What is the difference between the sum of nth powers and the sum of squares or cubes?

The sum of nth powers is a more general term, which includes the sum of squares and the sum of cubes as special cases. The sum of squares only involves the squares of numbers, while the sum of cubes only involves the cubes of numbers. The sum of nth powers involves any power of numbers, not just squares or cubes.

5. How do you find the expression for a sequence involving the sum of nth powers?

To find the expression for a sequence involving the sum of nth powers, you would first need to identify the pattern or rule for the sequence. Then, you can use algebraic methods to derive a formula or equation that represents the sum of the first n powers in the sequence. This may involve using the sum of geometric series, the binomial theorem, or other mathematical concepts.

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