Pascal's Triangle and Powers of 11Date: 02/21/99 at 23:01:18 From: Robert Chung Subject: Pascal's Triangle and Powers of 11 Hi there. I've been working on a project for my finite math class and was wondering if you could help me with a certain pattern found in Pascal's Triangle. There seems to be a relation between the rows of Pascal's Triangle and the Powers of 11. For example: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 70 1 The first row (1) is 11^0 = 1 The second row (1 1) is 11^1 = 11 The third row is (1 2 1) 11^2 = 121 And so on. However, the pattern becomes harder to see after the 6th row; i.e: 1 5 10 10 5 1 becomes 11^5 = 161051 A funny way to work this out is: 1 5 10 10 5 1 1 (5 + 1) (0 + 1) (0 + 5) 1 The rest of the rows are harder to get. I was just wondering if there's a special way to find the pattern in each row of Pascal's Triangle, and to see whether a pattern can be found at the 12th row (1 11 55 165 330 462 330 165 55 11 1). Thanks a bunch. -Rob- Date: 02/21/99 at 23:13:53 From: Doctor Schwa Subject: Re: Pascal's Triangle and Powers of 11 Indeed, Pascal's triangle gives the coefficients of (x+y)^n in general, so for example the 1 3 3 1 row tells you that (x+y)^3 = x^3 + 3x^2 y + 3xy^2 + y^3 If you let x = 10, and y = 1, then you have 11^3 = 10^3 + 3*10^2 * 1 + 3*10*1^2 + 1^3 and since all the powers of 1 are just 1, you get 1331. For later rows, when you have numbers in the triangle that are bigger than 10, they "carry" to the next row. So, if you use the 12th row as you have it, you can just add 1 11 55 165 330 462 330 165 55 11 1 where the numbers are lined up so the rightmost digit moves over one unit each time (because you have one fewer power of ten to multiply it by). I hope that helps clear things up for you! - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/ |
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