Question

B={0,3,8,15,24,35} write builder form​

Answer 1

Step-by-step explanation:

I've written your numbers in the top row of the following array

03385201572024920351120481320

Each of the rows below is formed by the difference of the elements above it (directly above minus the one to the left)

Notice that the third row stabilizes at 2 and the following (and all other) rows are 0. This gives us confidence to believe that this system is giving a reliable prediction. So we extend the second row to 13 and produce 48 as the first value not provided by your question and it should be clear how to proceed from here.

In general, if the k'th differences are constant, then the sequence can be represented by a degree k polynomial. To find out exactly which polynomial we employ Stirling numbers which are explained in the following Wkipedia article in more detail:

Without using this apparatus we still know the following:

The third row is represented by the polynomial

f_3(n)=2 (degree 0)

The second row is like this:

f_2(n) = 2n+1

f_3(n) = a n (n-1) + b n + c

Where f_3(0)= c = 0

f_3(1) = b + c = 3

f_3(2) = 2a + 2b + c = 8

From this information we can easily recover a, b, and get f_3(n) explicitly.

It's useful in problems of this kind to use base polynomials of the form

n(n-1)(n-2)

etc

and later expand them to the usual power format. Stirling numbers can be used to automate this.