SOLUTION: My math teacher has asked me to find the n^th term for the fourth diagonal in Pascal's Triangle. Basicly, he has asked me to find a formula that will find the next number for the f

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Question 159183: My math teacher has asked me to find the n^th term for the fourth diagonal in Pascal's Triangle. Basicly, he has asked me to find a formula that will find the next number for the fourth diagonal. I've tried all sorts of charts to derieve a formula but none have yeilded any.
Answer by gonzo(654) (Show Source):
You can put this solution on YOUR website!
the equation for the 3d diagonal is given as:

the equation for the 4th diagonal is given as:

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try these websites.
the first one describes pascal's triangle.
the second one actually give you equations to calculat the nth term.
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the first website is: http://www.mathsisfun.com/tetrahedral-number.html
the second website is: http://noticingnumbers.net/220PASCALStriangle.htm
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the formulas i provided you are taken from the second website and manipulated so the manual calculations are easier.
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a test of the formulas for the 9th term is constructed below.
the answer should be 45 for the 3d diagonal and 165 for the 4th diagonal.
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for the 3d diagonal, the formula for the 9th term (n = 9) yields
(10*9)/2 which equals 90/2 which equals 45 which is correct.
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for the 4th diagonal, the formula for the 9th term (n = 9) yields
(11*10*9)/6 = 990/6 = 165.
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these formulas actually work !!!!!
hopefully this is what you were looking for.