Q.Prove the following by using the principle of mathematical induction for all $n \in N$: $1.2 + 2.2^2 + 3.2^3 + \ldots + n.2^n = (n-1)2^{n+1} + 2$

Let $P(n)$ be the statement

Base case: For $n=1$,

So $P(1)$ is true.

Inductive step: Assume $P(k)$ is true for some $k\ge1$:

We must show the sum up to $(k+1)$ equals $k\cdot2^{k+2}+2$.

Adding the next term $(k+1)\cdot2^{k+1}$ to both sides of the hypothesis:

$=2^{k+1}\cdot2k+2=k\cdot2^{k+2}+2.$ …

Arithmetic-geometric-series induction. Add the next term $(k+1)2^{k+1}$ to the closed form, factor out the common $2^{k+1}$, and simpli …

• Adding $2^{k+1}$ instead of $(k+1)2^{k+1}$ as the next term (dropping the coefficient).
• Combining the powers of $2$ incorrectly, e.g. writing $2^{k+1}\cdot2k=k\cdot2^{k+1}$ instead of the correct …