Question Consider the Recurrence Relation anan12n with initi

Question) Consider the Recurrence Relation

a_n=a_n1+2n with initial condition a1=2

And use that to do the following problems:

Write down the first five terms

a1,a2,a3,a4, and a5

Use FORWARD-SUBSTITUTION (specifically, and NO OTHER METHOD) to guess a closed-form for the sequence.

Prove by induction that your closed form is correct. (Remember, to PROPERLY do this by induction, your proof MUST use both the recurrence relation AND the induction hypothesis at some point in your proof!)

Solution

Given:a_n=a_n1+2n

             and a1=2.

             a2=a1+2n

               =2+2(2)=6

            a3=a2+2n

=6+2(3)=12

   a4=a3+2n

    =12+2(4) =20

a5=a4+2n

   =20+(5) =30

     a1,a2,a3,a4,a5=2,6,12,20,30