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nth term in this formula..HELP! think im going crazy!?
Yes its true i have a headache!
Please help me over this...
This is my number sequence here:
Sum: 3 4 5 6 7
Trios: 1 3 6 10 15
Pattern: 2 3 4 5
2nd: 1 1 1
So above is my number sequence of how many trios can you find with the sum of 3,4,5,6, and 7.
the first difference shows that each one trio you add 1 and it goes in accending order, then the second difference shows that you constantly add 1.
So here my first difference is +2 +3 +4 +5 (changing difference sequence)
second: +1 +1 +1
the formula i was given was..
numbers of trios for sum (n) = (n-1) (n-2) / 2
so for example for the sum of 3..
n=3 (n-1) (n-2)/2 = (3-1) (3-2)/2=2x1/2=1
Is this right?? and if so please explain it for me.. i have searched on the internet and come across another formula, so not sure which is right?
Thank-u! x
Yes the formula that you have been given is in fact correct. It is a subtle variation on a standard result. Notice that the sequence that you have for the number of trios is actually the triangle numbers, or the sum of the first n natural numbers:
1 = 1
3 = 1 + 2
6 = 1 + 2 + 3
10 = 1 + 2 + 3 + 4
15 = 1 + 2 + 3 + 4 + 5
If we label these 1, 2, 3, 4, 5, then the nth term for these sequences is a well known result:
n(n+1)/2
However, your first term is actually 3, rather than 1, so instead of using 1 in the formula, you should use n - 2, since when n = 3, n - 2 = 1. So if you replace every n in the formula by n - 2, you get:
(n - 2)(n - 2 + 1)/2 = (n - 2)(n - 1)/2, which is the formula you were given.
Unfortunately I can't really help as to exactly why the number of trios comes up with this pattern, although obviously this pattern only appears if you include the permutations of each trio, ie. (1, 1, 2), (1, 2, 1) and (2, 1, 1) are permutations of each other. I'd only presume that this pattern has something has something to do with permutations.
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Tags: crochet, design, knitting, pattern, pattern triolet, sto triolic pattern enhancer, triolic pattern enhancer, visualization