6. Neil enjoys playing around with sequences in his spare time, and one day he decides to create a new one. He does this by taking a normal arithmetic sequence but repeatedly halving the common difference between the terms. For instance, if the difference between the first two terms is 16 , the difference between the third term and the second term will be 8 and the difference between the fourth term and the third term will be 4 (continuing in this fashion for as long as he likes).
\begin{enumerate}[label=(\alph*)]
\item If $u _ { 1 } = a$ and $u _ { 2 } = a + d$, find a closed form expression for $u _ { n }$, where $n \in \mathbb { N }$. "Closed form" means that you can't have a "..." or sigma notation in your final answer.
\item With $u _ { n }$ defined as above, find a closed form expression for $S _ { n } = \sum _ { k = 1 } ^ { n } u _ { k }$, where $n \in \mathbb { N }$.
\item With $u _ { n }$ and $S _ { n }$ defined as above, find the values of $a$ and $d$ if $u _ { 7 } = 23$ and $S _ { 7 } = 41$.\\[0pt]
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\end{enumerate}

\hfill \mbox{\textit{SPS SPS FM 2024 Q6 [10]}}