Consider the sequence a1 2 a2 5 a3 9 a414 etc a Find a re

Consider the sequence a₁ = 2, a₂ = 5, a₃ = 9, a₄=14, etc.

(a) Find a reoccurance relation that expresses a_n in terms of a_n-1 for every integer n >= 2.

(b) Conjecture an explicit formula for a_n, and then prove that your conjecture is correct.

Solution

1 = a2 - a1 = 5 - 2 = 3
d2 = a3 - a2 = 9 - 5 = 4
d3 = a4 - a3 = 14 - 9 = 5
and
d2 - d1 = d3 - d2 = 1
with constant differences after 2 subtractions, you know the polynomial function is quadratic, so

a(1)² + b(1) + c = 2 ........... a + b + c = 2
a(2)² + b(2) + c = 5 ........... 4a + 2b + c = 5
a(3)² + b(3) + c = 9 ........... 9a + 3b + c = 9

5a + b = 4
3a + b = 3
2a = 1, a = ½
3(½) + b = 3
b = 1½
c = 0

so a_n = (n² + 3n) / 2