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).

1. 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.
2. With \(u _ { n }\) defined as above, find a closed form expression for \(S _ { n } = \sum _ { k = 1 } ^ { n } u _ { k }\), where \(n \in \mathbb { N }\).
3. With \(u _ { n }\) and \(S _ { n }\) defined as above, find the values of \(a\) and \(d\) if \(u _ { 7 } = 23\) and \(S _ { 7 } = 41\).
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