2023 HCI CT1 Q5 [Modified]
(b)
A sequence ${{a}_{1}}$, ${{a}_{2}}$, ${{a}_{3}}$, … is given by
${{a}_{1}}=\frac{1}{2}$ and ${{a}_{n}}=\frac{{{n}^{2}}}{{{2}^{n}}}$ for $n\ge 1$.
Let ${{S}_{n}}$ be the sum of the first $n$ terms of the above sequence. Find the values of $p$ and $q$ such that ${{S}_{n}}=6-\frac{{{n}^{2}}+pn+q}{{{2}^{n}}}$ where $p$ and $q$ are constants.
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