Discrete Math Informal Proofs Using Mathematical Induction

Need to do a proof by mathematical induction using 4 steps to show that the following statement is true for every positive integer n and to help use the weak principle of mathematical induction.
$2 + 6 + 18 + ... + 2\times{3^{n-1}} = 3^n-1$

1. show that the base step is true




2. What is the inductive hypothesis?




3. what do we have to show?




4. proof proper (Justify each step):

Answer

Base Step: $2 \cdot 3^{1-1} = 2 = 3^1 - 1$

The inductive hypothesis is: $\sum_{n=1}^{k} 2 \cdot 3^{n-1} = 3^k - 1$

We must show that under the assumption of the inductive hypothesis that $$3^k - 1 + 2 \cdot 3^k = 3^{k + 1} - 1$$

We verify this as $$3^k - 1 + 2 \cdot 3^k = 3^k(1 + 2) - 1$$
$$= 3^{k+1} - 1$$