Do a proof by mathematical induction using the 4 steps below to show that the fo

Question

Do a proof by mathematical induction using the 4 steps below to show that the following statement is true for every positive integer n: 2 + 6 + 18 + ... + 2*3^(n-1) = (3^n)-1. use the weak principle of mathematical induction and state whether you are proving P(k) --> P(k+1) or P(k-1) --> P(k). 1)What is the base step. 2)what is the inductive hypothesis. 3)what do we have to show. 4)proof proper:

Explanation / Answer

4) Consists of everything below: 1) Base step: Let n=1. Observe that 2*3^(n-1)=2*3^0=2*1=2=(3^1)-1=3-1=2. 2) We are proving that P(k)->P(k+1), where P(n) is the statement such that 2+6+18+...+2*3^(n-1)=(3^n)-1. Then suppose P(k) holds. Then 2+6+18+...+2*3^(k-1)=(3^k-1). 3) Then observe that P(k+1) is the statement that we want to show, which is that 2+6+18+...+2*3^(k)=3^{k+1}-1. By assumption of P(k), we can see that (2+6+18+...+2*3^{k-1})+2*3^k=(3^k)-1+2*3^k=3(3^k)-1=3^{k+1}-1, as desired.

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