Q.Prove the following by using the principle of mathematical induction for all $n \in N$: $1.3 + 2.3^2 + 3.3^3 + \ldots + n.3^n = \dfrac{(2n-1)3^{n+1}+3}{4}$

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 $\dfrac{(2k+1)3^{k+2}+3}{4}$.

Adding the next term $(k+1)\cdot3^{k+1}$ to both sides of the hypothesis:

Simplify the bracket: …

Arithmetic-geometric-series induction. Add the next term $(k+1)3^{k+1}$ to the closed form, combine over the common denominator $4$, factor $3^{k+1}$ out of the numerat …

• Forgetting to convert $(k+1)3^{k+1}$ to a fraction with denominator $4$ (i.e. $\frac{4(k+1)3^{k+1}}{4}$) before combining with the hypothesis. …