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 ~ by ciceronii
Solution 2
Call the row , and the position of the terms . Call the middle term in the ratio . The first term is , and the final term is . Because we have the ratio ,
and .
and
Solve the equations to get and .
-Solution and LaTeX by jackshi2006, variables and algebra simplified by oinava
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