# How do you find the nth term formula for a sequence with a non-constant difference?

To calculate the nth term formula of a sequence with non-constant difference.
1. Calculate the difference between the terms.
2. Calculate the change in difference of the terms.

Let’s take the sequence

-3, 8, 23, 42, 65…

1. Calculate the difference between the terms

8 - (-3) = 11

23 - 8 = 15

42 - 23 = 19

65 - 42 = 23

The difference between the terms is not constant. Therefore, calculate the change in difference of the terms.

15 - 11 = 4

19 - 15 = 4

23 - 19 = 4

The change in difference between the terms is constant. As the second difference is constant, the sequence is quadratic i.e. an2 + bn +c.

a = second difference / 2

a = 4/2 = 2
2n2 + bn +c
n = 1, 2, 3, 4, 5
2. Calculate 2n2:
n = 1 , 2n2 = 2 (1)2 = 2
n = 2 , 2n2 = 2 (2)2 = 8
n = 3 , 2n2 = 2 (3)2 = 18
n = 4 , 2n2 = 2 (4)2 = 32
n = 5 , 2n2 = 2 (5)2 = 50
3. Subtract the answers of 2n2 from original sequence:
-3, 8, 23, 32, 65
2, 8, 18, 26, 50
-3 - 2 = -5
8 - 8 = 0
23 - 18 = 5
42 - 32 = 10
65 - 50 =15
4. Calculate the difference of the sequence -5, 0, 5, 10, 15:
0 - (-5) = 5
5 - 0 = 5
10 - 5 = 5
15 - 10 = 5
d = 5
5. Calculate a linear sequence using an = a + (n -1) d
a = first term = -5
d = 5
an = -5 + (n-1) 5
an = -5 + 5n -5
an = 5n -10
bn + c = 5n -10
b = 5
c = -10
The formula for sequence: 2n2 + 5n -10