It may be noted that for a mathematical sequence, we can find a generalized formula for the term. There's a theoretical approach, which works in most of the cases. It is based on simple algebra. It states:
If in a sequence, the difference comes out to be a constant, then the general formula for the term is a degree polynomial.
It is basically a converse of the method of finite differences , which states that if we make a sequence of numbers obtained by plugging consecutive numbers in a degree polynomial, then the difference will come out to be a constant.
So, let's understand this with an example.
Find the term in the sequence .
So, we start by taking consecutive differences, and keep on making newer and newer sequences named , and so on till we get a constant. Let's do it out.
Subtracting consecutive terms, like and so on, we have
Similarly, we have
YAY!! So came to be a constant, and so the general term of this sequence is a second degree polynomial. Now we shall find out which one it is.
Let the sequence's general term be defined as , where are real numbers. We know that , , and . So we have enough equations to solve and find , and .
We find that , and .
Kudos! We have generated the term. Now everything is easy. The term is , which is indeed .