Deriving the Closed-Form of \( S(n) = \sum_{k=0}^n k2^k \)
Exploring the Perturbation Method
Step 1
Adding the Last Term
We start by expressing \( S(n+1) \) by adding the next term in the sequence to the existing sum \( S(n) \). This provides our first perspective on the expanded sum.
\[ S(n+1) = S(n) + (n+1) 2^{n+1} \]
Step 2
Extracting the First Term
Alternatively we pull out the very first term where the index is zero. We then examine the remaining summation from \( k=1 \) up to \( k=n+1 \).
\[ S(n+1) = 0 \cdot 2^0 + \sum_{k=1}^{n+1} k 2^k \]
Step 3
Re-indexing and Substitution
We shift the index so it matches the limits of the original sum. This allows us to expand the terms and identify a standard geometric series component.
\begin{align*}
S(n+1) &= \sum_{k=0}^n (k+1)2^{k+1} \\
&= 2 \sum_{k=0}^n (k+1)2^k \\
&= 2 \left( \sum_{k=0}^n k2^k + \sum_{k=0}^n 2^k \right) \\
&= 2(S(n) + 2^{n+1} - 1)
\end{align*}
Step 4
Final Algebraic Solution
We equate the two expressions for the expanded sum and perform the final isolation of the unknown variable.
\begin{align*}
S(n) + (n+1)2^{n+1} &= 2S(n) + 2 \cdot 2^{n+1} - 2 \\
S(n) &= n \cdot 2^{n+1} - 2^{n+1} + 2 \\
S(n) &= (n-1)2^{n+1} + 2
\end{align*}
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