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.
Let us suppose a sequence, with no common difference.
Further taking out the difference between each term and writing that in the sequence form.
Since we have not got a common difference, we'll repeat the above step further again.
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.