Method of differences

Main post link -> https://brilliant.org/assessment/techniques-trainer/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.

Easy Math Editor

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`2 \times 3`	$2 \times 3$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

Are you talking about the values of $f(n)$ , or the closed form?

Tim Vermeulen - 7 years, 9 months ago

The original polynomial of the form $f(x) = ax^n + bx^{n-1} + \dots$

Priyansh Sangule - 7 years, 9 months ago

Okay. Could you give an example of such a table? I'd be happy to help you find the original polynomial.

Tim Vermeulen - 7 years, 9 months ago

@Tim Vermeulen – 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 $f(x) = ax^n + bx^{n-1} + \dots$

Please help and do elaborate !

Thank you

Priyansh Sangule - 7 years, 9 months ago

I'm encourage you to read through Worked Example 1, and do the (*) exercise, which gives you the answer.

Calvin Lin Staff - 7 years, 9 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!

Please help , Thank you !

Priyansh Sangule - 7 years, 9 months ago

@Sreejato Bhattacharya @Yan Yau Cheng @Aditya Raut @Tim Vermeulen @Bhargav Das @Calvin Lin

I need help guys :)

Priyansh Sangule - 6 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$