1992 AIME Problems/Problem 4
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 ?
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
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 .
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