Quadratic Patterns Question

Question

Consider the sequence below:

-2, 1, 6, 13, 22, …

What explicit expression can be used to find the nth term of this sequence?

Answer 1

a_n = an² + bn + c

n = 1; a + b + c = - 2 (1); (2) - (1): 3a + b = 3

n = 2; 4a + 2b + c = 1 (2) (3) - (2): 5a + b = 5

n = 3; 9a + 3b + c = 6 (3) 2a = 2; a = 1; b = 0, c = - 3

So, a_n = n² - 3.

You check for n = 4 and n = 5

Answer 2

-2,1,6,13,22,33,46,61,78

a(1) = -2 = -3 + 1

a(2) = 1 = -2 + 3

a(3) = 6= 1 + 5

a(4) = 13 = 6 + 7

a(5) = 22 = 13 + 9

a(6) = 33 = 22 + 11

a(7) = 46 = 33 + 13

a(8) = 61 = 46 + 15

a(9) = 78 = 61 + 17

78 is the 9th term

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

a(9) = a(8) + 2(9) + 1 = 68 +17

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

nth term = (n-1)th term plus (2n-1) or = previous term + 2n - 1