Difference between revisions of "1997 AJHSME Problems/Problem 19"
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As above, when <math>a=18</math>, <math>b=17</math>, and the sum is <math>35</math>, or <math>\boxed{D}</math> | As above, when <math>a=18</math>, <math>b=17</math>, and the sum is <math>35</math>, or <math>\boxed{D}</math> | ||
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== See also == | == See also == | ||
Latest revision as of 21:00, 21 April 2017
Contents
Problem
If the product 
, what is the sum of 
and 
?
Solution 1
Notice that the numerator of the first fraction cancels out the denominator of the second fraction, and the numerator of the second fraction cancels out the denominator of the third fraction, and so on.
The only numbers left will be in the numerator from the last fraction and 
in the denominator from the first fraction. (The will cancel with the numerator of the preceeding number.) Thus, 
, and 
.
Since the numerator is always one more than the denominator, 
, and 
, giving an answer of 
Solution 2
Find a pattern. If 
, then the expression is just the first two terms, which is .
If 
, then the expression is the first three terms, giving 
.
If 
, the expression is the first four terms, giving 
.
If 
, the expression will be the first five terms, giving 
.
Conjecture that the expression is always going to equal 
, and thus when , the expression will be 
, as desired.
As above, when , 
, and the sum is 
, or
See also
| 1997 AJHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 18 |
Followed by Problem 20 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
