Quadratic Patterns Question

Question

Consider the sequence below:

-9, -6, -1, 6, 15 …

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

Answer 1

The difference between each number is the next odd number.

Since this is looking for a quadratic sequence:

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

a(1) = x + 1 = -9 x=-10 --> a(1) = -10 +2n -1

a(2) = a(1) + 3 = -10 + 1 + 3 = -6

a(3) = a(2) + 5 = -10 +1 +3 + 5 = -1

a(4) = a(3) + 7 = -10 + 1 +3 + 5 +7 = -10 + 25 = 15

or

a(n) = -10 + ∑ⁿ_i=1(2i-1)

It can be shown that ∑ⁿ_i=1(2i-1) = n²

So the closed form is:

a(n) = n²-10

a(1) = 1-10 = -9

a(2) = 4-10 = -6

a(3) = 9-10 = -1

a(4) = 16-10 = 6

a(5) = 25-10 = 15

Answer 2

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

any term = the previous term +2n -1

15 = 6 +2(5) -1 = 15 = the 5th term

Answer 3

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

any term = the previous term +2n -1

15 = 6 +2(5) -1 = 15 = the 5th term