A sequence {an} is defined recursively with recurrence relation an = 2an1 an2 for n 3

Problem 6

A sequence  is defined recursively with recurrence relations  for .

(a) For  and  

         i. determine  and

         ii.  conjecture a formula for  for each positive integer n.

         iii.  verify the conjecture in

(b) Repeat (a) for  and =2

(c) Repeat (a) for  and =1

                                                        Step by Step Solution

Step 1 of 3

(a)

Consider the given sequence is,

 for

Given that  and

Solve for,

Solve for,

Solve for,

(ii)

The part (i) shows that the sequence is 1, 3, 5, 7, 9.

The common difference of the sequence is 2.

Then, by the general equation, the formula for the nth term of the sequence is,

(iii)

The formula obtained from part 2 is .

The first term verification is,

The second term verification is,

The second term verification is,

The second term verification is,

The second term verification is,

Hence, the conjecture is verified.