For each of the following closed forms write out the first s

For each of the following closed forms, write out the first several terms of the sequence (at least five) and use this to create a recurrence for the sequence.

If possible, please write out how you got the recurrence.

Solution

a) a_n = 3n + 1

    a_(n+1) - a_n = 3(n+ 1) +1 -3n - 1 = 3

    a1 = 3x1+ 1 = 4, a2 = 3 x 2 + 1 = 7, a3 = 3 x 3+ 1 = 10 and so on...

Recurrence relation: a_(n+1) = a_n + 3

b) a_n = 2n + 7

    a_(n+1) - a_n = 2(n+ 1) +7 -2n - 7 = 2

    a1 = 2x1+7 = 9 , a2 = 2x2+7 = 11, a3 = 2x3+7 = 13

Recurrence relation: a_(n+1) = a_n + 2

c)a_n = 3n² + n

   a_(n+1) = 3(n+1)² + (n+1) = 3n² + 6n + 3 + n + 1 = 3n² + 7n + 4

a1 = 4, a2 = 14 a3 = 30

Recurrence relation:

a_(n+1) - a_n = 3n² + 7n + 4 - 3n² - n = 6n + 4 = 2(3n+2)

a_(n+1) = a_n + 2(3n+2)

d)a_n = 2^(2n-1) -1

a1 = 1 , a2 = 7, a3 = 31

Recurrence relation:

    a_(n+1) = 2^(2(n+1)-1) - 1 = 2⁽²ⁿ⁺¹⁾ - 1

   a_{(n+1) -} a_n = 2⁽²ⁿ⁺¹⁾ - 1 -(2^(2n-1) -1) =2⁽²ⁿ⁺¹⁾ - 2^(2n-1) = 2²ⁿ (2 - (1/2)) = 3.2^(2n-1)

a_(n+1) = a_n + 3.2^(2n-1)