Difference between revisions of "1992 AIME Problems/Problem 4"
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== Problem == | == 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 <math>3: 4: 5</math>? | + | 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 <math>3: 4: 5</math>? |
== Solution == | == Solution == | ||
Revision as of 15:28, 2 March 2008
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
In Pascal's Triangle, we know that the binomial coefficients of the 
th row are 
. Let our row be the th row such that the three consecutive entries are 
, 
and 
.
After expanding and dividing one entry by another (to clean up the factorials), we see that 
and 
. Solving, 
.
| 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 | ||
