Difference between revisions of "1992 AIME Problems/Problem 4"
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Call the row x, and the number from the leftmost side t. Call the first term in the ratio N. This is <math>\dbinom{x}{t}</math>. The next term is <math>N * \frac{x-t}{t+1}</math>, and the final term is <math>N * \frac{(x-t)*(x-t-1)}{(t+1)*(t+2)}</math>. Because we have the 3:4:5 ratio, <math>\frac{x-t}{t+1} = \frac{4}{3}</math> and <math>\frac{(x-t)*(x-t-1)}{(t+1)*(t+2)} = \frac{5}{3}</math>. | Call the row x, and the number from the leftmost side t. Call the first term in the ratio N. This is <math>\dbinom{x}{t}</math>. The next term is <math>N * \frac{x-t}{t+1}</math>, and the final term is <math>N * \frac{(x-t)*(x-t-1)}{(t+1)*(t+2)}</math>. Because we have the 3:4:5 ratio, <math>\frac{x-t}{t+1} = \frac{4}{3}</math> and <math>\frac{(x-t)*(x-t-1)}{(t+1)*(t+2)} = \frac{5}{3}</math>. | ||
Solve the equation to get get <math> t= 26 </math> and <math>x = \boxed{062}</math>. | Solve the equation to get get <math> t= 26 </math> and <math>x = \boxed{062}</math>. | ||
+ | https://www.wolframalpha.com/input/?i=%28x-t%29%2F%28t%2B1%29++%3D+4%2F3%2C++%28x-t%29%28x-t-1%29%2F%28%28t%2B1%29%28t%2B2%29%29+%3D+5%2F3 | ||
Revision as of 12:13, 24 August 2020
Problem
In Pascal's Triangle, each entry is the sum of the two entries above it. In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio ?
Solution 1
Consider what the ratio means. Since we know that they are consecutive terms, we can say
Taking the first part, and using our expression for choose , Then, we can use the second part of the equation. Since we know we can plug this in, giving us We can also evaluate for , and find that Since we want , however, our final answer is ~LaTeX by ciceronii
Solution 2
Call the row x, and the number from the leftmost side t. Call the first term in the ratio N. This is . The next term is , and the final term is . Because we have the 3:4:5 ratio, and . Solve the equation to get get and . https://www.wolframalpha.com/input/?i=%28x-t%29%2F%28t%2B1%29++%3D+4%2F3%2C++%28x-t%29%28x-t-1%29%2F%28%28t%2B1%29%28t%2B2%29%29+%3D+5%2F3
-jackshi2006
1992 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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