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\(f(n) = k^n\) for some \(n\)?

To be specific, there have been questions on this topic:

n = 0 to 5 (inclusive)

n = 0 to 6 (inclusive)

This wiki contains spoliers about these questions, so you are recommended to do them first before reading the rest of this note.

In these questions: Actually, please do those questions first.

In these questions, \(k=2\) and \(n = 0\) to \(n=5\) and \(n=0\) to \(n=6\).

There's a lot of similarities, and even the answer is amazing, but I shall not spoil it.

Both questions are in the form "\(f(n)\) is a \(p\)-degree polynomial. When \(q = 0,1,2,\ldots,p\), \( f(q) = k^q\). What is \(f(x)\)?"

These questions are a special case of this form, with \(k = 2\) and x = 2*q+1. The answer is also very similar, f(x) = 2^(2*q).

I would like to know if this can be generalised for q and also, k, and if so, what values of x can create these values.

I'll work on it when I have time. Which doesn't come a lot.

Note by Aloysius Ng
6 months, 4 weeks ago

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It can be done.

\({ 3 }^{ n }={ 2 }^{ 0 }\left( \begin{matrix} n \\ 0 \end{matrix} \right) +{ 2 }^{ 1 }\left( \begin{matrix} n \\ 1 \end{matrix} \right) +{ 2 }^{ 2 }\left( \begin{matrix} n \\ 2 \end{matrix} \right) +\cdots +{ 2 }^{ n-1 }\left( \begin{matrix} n \\ n-1 \end{matrix} \right) +2^{ n }\left( \begin{matrix} n \\ n \end{matrix} \right) \) Joel Yip · 5 months, 3 weeks ago

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