Question: Let (T(n)=sum_{k=0}^{n}(-1)^{k}left(begin{array}{l}n kend{array}ight)). (a) Use Pascal's Triangle to compute (T(n)) for (n=1,2,3,4). (b) Prove that (T(n)=0) for all (n geq 1). Expand ((a+b)^{n}) and
Let \(T(n)=\sum_{k=0}^{n}(-1)^{k}\left(\begin{array}{l}n \\ k\end{array}ight)\).
(a) Use Pascal's Triangle to compute \(T(n)\) for \(n=1,2,3,4\).
(b) Prove that \(T(n)=0\) for all \(n \geq 1\). Expand \((a+b)^{n}\) and evaluate at \(a=1, b=-1\).
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