Question 1117535: 0, 1, 4, 10, 20, 35….
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52788)   (Show Source):
You can put this solution on YOUR website! .
For your info:
The posts without question/questions are the subject of ignoring at this forum.
I personally consider such posts as demonstration of disrespect to the tutors.
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
"Disrespectful to the tutors" is a little harsh... but it WOULD be courteous for you to take the time to ask the question you have about the sequence.
Assuming the question is about the next term, or the next few terms, or even a formula for the n-th term in the sequence, there is a very nice pattern in these numbers which allows us to find a POSSIBLE answer.
You can see the pattern by looking at the differences between successive terms (the "first differences"), then looking at the differences between the first differences (the "second differences"), and then looking at the third differences.
0 1 4 10 20 35
1 3 6 10 15
2 3 4 5
1 1 1
When a row of differences is constant, you can find more terms of the original sequence by continuing the constant difference and working back up the array of numbers. For example, to find the next 3 numbers in the sequence, we can add 3 more 1's in the last row:
0 1 4 10 20 35 56 84 120
1 3 6 10 15 21 28 36
2 3 4 5 6 7 8
1 1 1 1 1 1
If the question was to find a formula for the n-th term in the sequence, this pattern makes it easy, because all the numbers in the array are elements of Pascal's Triangle. And in fact the formula for the n-th term in the sequence is
 :
t(1) = {2*1*0)/6 = 0
t(2) = (3*2*1)/6 = 1
t(3) = (4*3*2)/6 = 4
t(4) = (5*4*3)/6 = 10
etc....
AND NOW, having done all that work on this interesting sequence....
While this sequence has an interesting pattern which gives a very nice POSSIBLE solution to the problem, in fact IT IS IMPOSSIBLE to know the "right" answer to ANY problem like this.
ANY next numbers will form a valid sequence....
So, in the end, even if you HAD asked a question about this sequence, we would not be able to show you how to get THE answer.
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