To calculate the nth term formula of a sequence with non-constant difference.

- Calculate the difference between the terms.
- 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. an ^{2} + bn +c.**

**a = second difference / 2**

a = 4/2 = 2

2n^{2} + bn +c

n = 1, 2, 3, 4, 5

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, 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 a_{n} = a + (n -1) d

a = first term = -5

d = 5

a_{n} = -5 + (n-1) 5

a_{n} = -5 + 5n -5

a_{n} = 5n -10

bn + c = 5n -10

b = 5

c = -10

The formula for sequence: **2n**^{2} + 5n -10