2 75 points We will use Mathematical Induction to prove that

2. (75 points) We will use Mathematical Induction to prove that for n > = 1 (a) Prove the base case. (b) What is the Inductive Hypothesis? (c) Show the Inductive Step.

Solution

Given:

       

    a)base case: Let take i=1 n=2

     L.H.S=[5(1)-2]+[5(2)-2]    if n=1 then LHS=3

              =3+8=11

   R.H.S=(5n^2+n)/2

            =(5*4+2)/2=11

b)Simple example
3 + 11 + - - - - + 5n-2= [(5n² +n )/2] prove by inductive

assume it is true ( hypothesis ) and then

If n= n+1 the sum should be [(5(n+1)² +(n+1) )/2] prove it

The term of n+1 = 5(n+1)-2

The sum of (n+1) = [(5n² +n)/2] +5(n+1)-2

[(5n² +n)/2] +5(n+1)-2

[(5n²/2 +n/2]+5n+5-2

[(5n²/2 +n/2]+5n+3 => [(5(n+1)² +(n+1) )/2]

the hypothesis is true

c) for k value:

                 [ [(5(k)² +(k) )/2] it is true

   and for k+1

               [(5(k+1)² +(k+1) )/2] is also true.

this is inductive step.