Derivation of 2nd divided difference

[roldy]

I'm trying to understand how the second divided difference is formulated. I understand that the first divided difference is just the equation of a slope.

$f(x_{i},x_{i+1})=\frac{f(x_{i})-f(x_{i+1})}{x_{i}-x_{i+1}}$

Every source that I have read always jumps to the second divided difference by saying "and by induction"

$f(x_{i},x_{i+1},x_{i+2})=\frac{f(x_{i},x_{i+1})-f(x_{i+1},x_{i+2})}{x_{i+2}-x_{i+1}}$

How is induction used to get this equation?

 

[pari777]

http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-KANPUR/mathematics-2/node112.html

you can check the above site if you think it would be useful.

 

[HallsofIvy]

Having got two consecutive first differences,
$$\frac{f(x_i)- f(x_{i+1}}{x_i- x_{i+1}}$$
$$\frac{f(x_{i+1}- f(x_{i+2}}{x_{i+1}- x_{i+2}}$$

Now, for the first difference of those:
$$\frac{\frac{f(x_i)- f(x_{i+1}}{x_i- x_{i+1}}- \frac{f(x_{i+1}- f(x_{i+2}}{x_{i+1}- x_{i+2}}}{x_i- x_{i+2}}$$

 

[roldy]

Thank you. I new it was some type of substitution but I failed to think of using the first difference with the first differences.

 

[blue_raver22]

Induction is a mathematical proof technique that involves proving a statement for a base case and then showing that if the statement holds for a particular case, it also holds for the next case. In the case of the second divided difference, we can use induction to show that the formula holds for any number of points.

To understand this, let's look at the first divided difference formula again:

f(x_{i},x_{i+1})=\frac{f(x_{i})-f(x_{i+1})}{x_{i}-x_{i+1}}

This formula calculates the slope between two points, x_i and x_{i+1}. Now, let's add another point, x_{i+2}, to the equation:

f(x_{i},x_{i+1},x_{i+2})=\frac{f(x_{i},x_{i+1})-f(x_{i+1},x_{i+2})}{x_{i+2}-x_{i+1}}

We can see that the second divided difference is simply the first divided difference between two divided differences (f(x_{i},x_{i+1}) and f(x_{i+1},x_{i+2})) divided by the difference between the last two points (x_{i+2}-x_{i+1}). This is essentially the same as calculating the slope between the first divided difference and the third point.

Now, to use induction, we can start with the base case of two points and show that the formula holds. Then, we can assume that the formula holds for n points and show that it also holds for n+1 points. This will prove that the formula holds for any number of points, including the second divided difference.

In conclusion, induction is used to show that the formula for the second divided difference holds for any number of points by building on the formula for the first divided difference. This allows us to generalize the formula and use it for any number of points in a sequence.