How to find the nth term in a sequence with no constant difference?

If the sequence doesn't have the constant difference in its 1st level, it is bound to have a constant difference at the 2nd or 3rd level.

Answer: The given series has the nth term as n² + 3.

The first step is always to look at the difference between the terms, and keep searching for the level which provides the constant difference between each adjacent term.

Explanation:

Let us suppose a sequence, with no common difference.

4,7,12,19,28,...,.....,....

Further taking out the difference between each term and writing that in the sequence form.

3,5,7,9,.......

Since we have not got a common difference, we'll repeat the above step further again.

2,2,2,.......

Now since a common difference of 2 is received we can understand the nature of the series now. We need to note that we go on to subdivide the series, till we get a common difference.

In the above example since after two steps we got the unique common difference, therefore, the sequence of this series is quadratic, i.e of the form an² + bn + c

Keeping n = 1, a + b + c = 1st term = 4

for n = 2, 4a + 2b + c = 7

for n = 3, 9a + 3b + c = 12

To calculate the values of a, b, c; we can make use of the common difference.

==> 7 - 4 = (2nd term - 1st term) = 4a + 2b + c - a - b - c

==> 3 = 3a + b------(1)

==> 12 - 7 = (3rd term - 2nd term) = 9a + 3b + c - 4a - 2b - c

==> 5 = 5a + b--------(2)

On solving equation 1 and equation 2 we got the value for a and b and using the same value into 1st term, we obtain all three values.

a = 1, b = 0, c = 3

Quadratic equation = an² + bn + c = n² + 3 

Therefore, the nth term of the series, in this case, is n² + 3.