Sum with an exponential term
Task number: 3313
Prove via mathematical induction: \(\displaystyle\sum_{i=1}^n i2^i = (n-1)2^{n+1}+2\).
Solution
For \(n=1\):
\(\displaystyle\sum_{i=1}^1 1{\cdot} 2^1 = 2 = (1-1)2^{1+1}+2\).Induction step \(n-1 \to n\):
\(\displaystyle\sum_{i=1}^n i2^i = \left(\sum_{i=1}^{n-1} i2^i \right) + n2^n = (n-2)2^{n}+2 + n2^n = 2(n-1)2^{n}+2 = (n-1)2^{n+1}+2\).
