Give a recursive defintion of the sequence {an} n (1,2,3...) if?

Question

a) an= 4n-2
b)an =1+(-1)^n
c)an=n(n+1)
d)an=n^2

Answer 1

a) a0 = 4*0 - 2

a(n+1)-a(n) = 4(n+1) - 2 - (4n - 2) = 4n + 4 -n - 4n + 2 = 4

or a(n+1) = a(n) + 4

b) a0 = 1 + (-1)^0 = 1 + 1 = 2

a(n + 1) = 1+ (-1)^(n + 1) = 1 + (-1)^n * (-1) = 1 - (-1)^n =
= 1 - [1 + (-1)^n - 1] = 1 -[a(n) - 1] = 1 - a(n) + 1 = 2 - a(n)

c) a0 = 0*1 = 0

a(n + 1) = (n + 1)(n + 2) = n^2 + 3n + 2 = n^2 +n + 2n + 2 =
= n(n + 1) + 2(n + 1) = a(n) + 2(n + 1)

d) a(0) = 0

a(n + 1) = (n + 1)^2 = n^2 + 2n + 1 = a(n) + 2n + 1