Question
The procedure of proof should follow the following format. The answer should like that. Example Consider the sequence a1, a2, a3,... given by a1= 2,
The procedure of proof should follow the following format. The answer should like that.
Example
Consider the sequence a1, a2, a3,... given by a1= 2, a2= 5, and ai= ai-1+ 2ai-2for all i3.
Prove that, for all i1:ai= (1/3)(72i-1+ (-1)i)
Proof: We prove the statement by induction on i.
Base Case:
We prove that ai= (1/3)(72i-1+ (-1)i) when i=1 and i=2.
When i=1: we are given that a1= 2,and we see that (1/3)(72i-1+ (-1)i) = (1/3)(7-1) = 2, as required.
When i=2: we are given that a2= 5,and we see that (1/3)(72i-1+ (-1)i) = (1/3)(72+1) = (1/3)15 = 5, as require
Inductive step: We prove that, for all k2,if aj= (1/3)(72j-1+ (-1)j) for all j{1,...,k},then ak+1= (1/3)(72k+ (-1)k+1).
(1)Let k be an arbitrary positive integer such that k2.Assume thataj= (1/3)(72j-1+ (-1)j) for all j{1,...,k}.
(2)Since k2, we know that k+13, so the given recursive formula states that ak+1= ak+ 2ak-1.
(3)From (1), we know that ak= (1/3)(72k-1+ (-1)k) and ak-1= (1/3)(72k-2+ (-1)k-1).
(4)From (2) and (3), it follows that ak+1= (1/3)(72k-1+ (-1)k) + 2(1/3)(72k-2+ (-1)k-1)= (1/3)(72k-1+ (-1)k) +(1/3)(72k-1+ 2(-1)k-1)= (1/3)(72k-1+ (-1)k) +(1/3)(72k-1-2(-1)k)= (1/3)(72k-1+ 72k-1+ (-1)k- 2(-1)k)= (1/3)(72k- (-1)k) = (1/3)(72k+ (-1)k+1) , which completes the inductive step.
7. Prove by induction on n: for all integers n 0, for every set S containing exactly n elements, [P(S)2Step by Step Solution
There are 3 Steps involved in it
Step: 1
Get Instant Access to Expert-Tailored Solutions
See step-by-step solutions with expert insights and AI powered tools for academic success
Step: 2
Step: 3
Ace Your Homework with AI
Get the answers you need in no time with our AI-driven, step-by-step assistance
Get Started