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# Method of differences

Hello everyone ,

I've just learnt the "Method of differences" and I'm really fascinated by the method !

Can somebody help me out how can we find the original polynomial $$f(x)$$ ,

[ For example $$f(x)$$ of the form $$f(x)= ax^n + bx^{n-1} + \dots$$ ]

after drawing the difference table (i.e. after finding the values of $$f(a) , D_1(a) \dots$$ ) ?

Do elaborate 'cause I don't know much about Binomials .

Thank you.

Note by Priyansh Sangule
3 years, 4 months ago

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I'm encourage you to read through Worked Example 1, and do the (*) exercise, which gives you the answer. Staff · 3 years, 4 months ago

Yes sir, I did note that but couldn't it be written in a simpler form ? I saw some people using combinatorics for that!

Are you talking about the values of $$f(n)$$, or the closed form? · 3 years, 4 months ago

The original polynomial of the form $$f(x) = ax^n + bx^{n-1} + \dots$$ · 3 years, 4 months ago

Okay. Could you give an example of such a table? I'd be happy to help you find the original polynomial. · 3 years, 4 months ago

For example -

A cubic polynomial $$f(x)$$ has the following values -

$$f(1)=13 ; f(2)=32 ; f(3)=69 ; f(4)=130$$

Find $$f(x)$$ .

So for this , first of all , I draw a difference table

$$\begin{matrix} n & f(n) & D_1(n) & D_2(n) & D_3(n) & \dots \\ 1 & 13 & 19 & 18 & 6 & \dots \\ 2 & 32 & 37 & 24 & \dots \\ 3 & 69 & 61 & \dots \\ 4 & 130 & \dots \\ \vdots \end{matrix}$$

Now , at this point I run into trouble - that how can I find the polynomial of the form $$f(x) = ax^n + bx^{n-1} + \dots$$

Thank you · 3 years, 4 months ago