General Term Pattern Recognition

It may be noted that for a mathematical sequence, we can find a generalized formula for the $n^{th}$ 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 $n^{th}$ difference comes out to be a constant, then the general formula for the $n^{th}$ term is a $n^{th}$ 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 $n^{th}$ degree polynomial, then the $n^{th}$ difference will come out to be a constant.

So, let's understand this with an example.

Find the $8^{\text{th}}$ term in the sequence $1, 4, 9, 16, 25, ......$.

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So, we start by taking consecutive differences, and keep on making newer and newer sequences named $D_1$, $D_2$ and so on till we get a constant. Let's do it out.

$S=1, 4, 9, 16, 25, \ldots.$

Subtracting consecutive terms, $D_i=S_{i+1}-S_i$ like $d_1=3$ and so on, we have

$D_1=3, 5, 7, 9, \ldots.$

Similarly, we have $D_2=2, 2, 2, \ldots.$

YAY!! So $D_2$ came to be a constant, and so the general $n^{th}$ 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 $t_n=An^2+Bn+C$, where $A, B, C$ are real numbers. We know that $t_1=1$, $t_2=4$, and $t_3=9$. So we have enough equations to solve and find $A$, $B$ and $C$.

We find that $A=1$, $B=0$ and $C=0$.

Kudos! We have generated the $n^{th}$ term. Now everything is easy. The $8^{th}$ term is $8^2$, which is indeed $64$. $_\square$