# Sequence - inhomogeneous recursion

• MHB
• goohu
In summary, the person is trying to find a way to solve a sequence problem, but is having trouble because of the alternating term. They are helped by someone who explains a simpler way to solve the problem. They solve the problem by using a web calculator and then try to do it manually the next time.
goohu

I need some help with this task. My theory book only shows examples of how to solve sequences in the form :

𝑎𝑘 = A * 𝑎(𝑘−1) − B * 𝑎(𝑘−2).

But I've no idea how to solve this task because of the alternating term. I've included the Answer (called "Svar") to the task.

goohu said:
I need some help with this task. My theory book only shows examples of how to solve sequences in the form :

𝑎𝑘 = A * 𝑎(𝑘−1) − B * 𝑎(𝑘−2).

But I've no idea how to solve this task because of the alternating term. I've included the Answer (called "Svar") to the task.
One way to do this would be to replace $a_k$ by $b_k = a_k + c(-1)^k$ (where $c$ is a constant to be chosen later). Then $a_k = b_k - c(-1)^k$, and the recurrence equation for $a_k$ becomes $$b_k - c(-1)^k = 3(b_{k-1} - c(-1)^{k-1}) - (b_{k-2} - c(-1)^{k-2}) - 2(-1)^k,$$ $$b_k = 3b_{k-1} - b_{k-2} + (-1)^k(c + 3c + c - 2).$$ Now choose $c$ so that $5c-2=0$ (so $c = \frac25$). That eliminates the awkward $(-1)^k$ term from the $b_k$ equation, which you should now be able to solve. Having found the answer for $b_k$, you then have $a_k = b_k - \frac25(-1)^k$.

Thanks, that was a pretty solution! However the calculations got a bit messy while solving the characteristic equation for bk by hand so I went ahead and used a web calculator for it.

I'm going to give it another shot tomorrow solving it by hand.

We are not allowed to use a pocket calculator at the exam plus you lose a lot of credits if you go wrong somewhere in the calculations. That makes me a really angry student.

Edit: Solved the problem now! thanks again for the elegant solution.

Last edited:

## What is the definition of sequence inhomogeneous recursion?

Sequence inhomogeneous recursion is a mathematical concept that involves recursively generating a sequence of values where each value is dependent on the previous values in the sequence and also on an external factor or variable.

## How does sequence inhomogeneous recursion differ from regular recursion?

Regular recursion involves a function calling itself repeatedly to solve a problem, while sequence inhomogeneous recursion involves generating a sequence of values based on a recursive formula that takes into account both the previous values in the sequence and an external factor.

## What is the purpose of using sequence inhomogeneous recursion?

Sequence inhomogeneous recursion can be used to model and solve complex problems that involve a combination of recursive and non-recursive factors. It is often used in mathematical and scientific fields to describe and analyze real-world phenomena.

## What are some common examples of sequence inhomogeneous recursion?

One example is the Fibonacci sequence, where each number is the sum of the two previous numbers in the sequence. Another example is the population growth of a species, where the population in each generation depends on the previous generations and environmental factors.

## What are the limitations of using sequence inhomogeneous recursion?

Sequence inhomogeneous recursion can become computationally expensive and time-consuming for large and complex problems. It also requires a clear understanding of the recursive formula and external factors, as well as careful handling of boundary conditions to avoid infinite loops.

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