1992 AIME Problems/Problem 4
Problem
In Pascal's Triangle, each entry is the sum of the two entries above it. The first few rows of the triangle are shown below.
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 . . The next term is , and the final term is . Because we have the ratio, = ,
and . Solve the equation to get get and .
-Solution and LaTeX by jackshi2006
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