I am give $s_n = 2^n + 3^n + 4^n$ for $n = 1,2,...$
so far I have
n = 1 : 2 + 3 + 4 = 9
n = 2 : 4 + 9 + 16 = 29
n = 3 : 8 + 27 + 64 = 99
n = 4 : 16 + 81 + 256 = 353
I am having a hard time trying to figure out the pattern.
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1The pattern is that they are $2^n+3^n+4^n$. Do you need something more than that, and if so what? – Angina Seng Oct 13 '17 at 14:28
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I need to find the recursive relation that satisfies $s_n = 2^n + 3^n +4^n$ – Bart Scott Oct 13 '17 at 14:29
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Well, I think the question is asking for $s_n$ as a function of preceding terms? – Oct 13 '17 at 14:30
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$s_n = s_{n-1} + 2^{n-1} + 2 \cdot 3^{n-1} + 3 \cdot 4^{n-1}$? – Randall Oct 13 '17 at 14:31
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The characteristic equation must be \begin{eqnarray*} (x-2)(x-3)(x-4)=0. \\ \end{eqnarray*} So the recurrence relation must be \begin{eqnarray*} \color{blue}{s_{n+3}-9s_{n+2}+26s_{n+1}-24s_n=0}. \\ \end{eqnarray*}
Donald Splutterwit
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