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{\Large NMU Math \& CS Department \\
\Large Problem of the Month \\
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October 2023 }
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\paragraph{} Consider the sequence $(a_{n})$ with initial conditions $a_{1}=1$, $a_{2}=2$, and recursion given by:

$$a_{n}=4a_{n-1}-3a_{n-2} \hspace{3mm} n \geq 3.$$

\paragraph{} The first few terms of the sequence are $1,2,5,14,41,122,\dots$ .

\paragraph{} Let $f(n)=a_1+a_2+ \cdots +a_n$ be the function that computes the sum of the first $n$ terms of the sequence $(a_n)$. For example: $f(1)=a_1=1$, $f(2)=a_1+a_2=3$, $f(3)=a_1+a_2+a_3=8$.

\paragraph{} Find a closed formula for $f(n)$.

\paragraph{} \textit{A closed formula for $f(n)$ is an expression that computes the value of $f(n)$ only using the value of $n$. Something like $f(n)=3n^2-2$ would be considered a closed formula, but $f(n)=a_{n-1}+n^2-3$ would not be considered a closed formula, because it relies on the value of a prior term in the sequence.}

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