Please solve this problem and SHOW AS MUCH DETAIL AS POSSIBLE. Also, if you need

Question

Please solve this problem and SHOW AS MUCH DETAIL AS POSSIBLE. Also, if you need to hand write the answer PLEASE WRITE AS NEATLY AS POSSIBLE. This will help me to really understand how to to the problem. Thank you so much for your help.

Problem 1: a) User iteration to guess an explicit formula for the sequence below defined recursively. Use the sequence formulas to simplify your answer whenever possible. (b) Use mathematical induction to verify that the formula of part (a) is correct. ck 3c 1, for all integers k 2 k-1 C1 1

Explanation / Answer

c1=1

c2=3c1 + 1 = 3+1 = 4

c3 = 3c2 + 1 = 12+1 = 13

c4 = 3c3 + 1 = 39+1 = 40

c5 = 120+1=121 and so on.

Thus, the last line has consecutive powers of 3. Thus, we can write an as:

an = 1+3+32 + ..... + 3n-1.

Thus recursively, an can be written as :

a1=1

an = an-1 + 3n-1 if n>1.

b) Using mathematical induction to prove my answer:

Let, P(n) = an-1 + 3n-1

First, we have to prove that the recursion holds true for n=1.

Thus, a1 = a0 + 30 = 1(Since, a0 = 0).

Thus, for n=1, the equation holds true.

Let us assume that the equation is true for n=k.

Thus,

P(k) = ak-1 + 3k-1.

Now, we have to prove that P(n) is true for n = k+1 as well.

P(k+1) = ak+1-1 + 3k+1-1 = ak + 3k which is true.

Thus, as the equation is true for n=k+1, thus, P(n) is correct and verfied using mathematical induction.

n 1 2 3 4 5 6 an 1 4 13 40 121 364 an - an-1 3 9 27 81 243

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