jkwiens.com: Answer: General Form of Sequence

Answer: General Form of Sequence

Saturday, 1 of March , 2008 at 9:59 pm

Question: The solution to $x = \sqrt{10y+1}$ where $x,y \in \mathbb N$ yields the following sequence:

$8,~12,~36,~44,~84,~96,~152,~168,~240~\ldots$

Find the general form for this sequence.

Answer: When analyzing this sequence, the first thing I tried was comparing the difference between each number in the sequence.

Lets investigate this new sequence.

$4,~24,~8,~40,~12,~56,~16,~72~\ldots$

It appears the odd terms are related to each other and so are the even terms. Lets break the odd terms and even terms into two different sequences.

$odd~terms:~4,~8,~12,~16,\ldots$

$even~terms:~24,~40,~56,~72,\ldots$

As you can see, the odd terms has the form:

$\phi_n = 4n$

The even terms, likewise, can be seen to have the form:

$\phi_n = \phi_{n-1} + 16$ or

$\phi_n = 16n + 8$

If we merge these results, we can get pattern for the following sequence

$4,~24,~8,~40,~12,~56,~16,~72~\ldots$

which is

$\phi_n = \begin{cases}4(\frac{n+1}{2}) & odd~n \\ 16(\frac{n}{2}) +8 & even~n\end{cases}$

$\phi_n = \begin{cases}2(n+1) & odd~n \\ 8(n+1) & even~n\end{cases}$

We can then use this result to find the solution to

$8,~12,~36,~44,~84,~96,~152,~168,~240~\ldots$

which would be

$\psi_n = \psi_{n-1} + \phi_n$

$\psi_n= \psi_{n-1} + \begin{cases}2(n+1) & odd~n \\ 8(n+1) & even~n\end{cases}$

where $\psi_0 = 8$ .

Category: Algebra, Answers

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