Difference between revisions of "1989 AHSME Problems/Problem 10"
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== Solution == | == Solution == | ||
| + | Repeatedly applying the function, and simplifying, we get <cmath>a,\quad-\frac1{a+1},\quad-\frac{a+1}a,</cmath>and then <math>a</math> again. So <math>a</math> must appear at every third term after <math>u_1</math>. The only option given of the form <math>1+3k</math> is <math>\boxed{\mathrm{(C)}\,16}</math>. | ||
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== See also == | == See also == | ||
{{AHSME box|year=1989|num-b=9|num-a=11}} | {{AHSME box|year=1989|num-b=9|num-a=11}} | ||
Latest revision as of 12:48, 3 February 2016
Problem
Consider the sequence defined recursively by 
(any positive number), and 
, 
For which of the following values of 
must 
?
Solution
Repeatedly applying the function, and simplifying, we get ![\[a,\quad-\frac1{a+1},\quad-\frac{a+1}a,\]](http://latex.artofproblemsolving.com/e/a/c/eac1570c9c46b33133c71b978c29a848e6864565.png)
and then 
again. So must appear at every third term after 
. The only option given of the form 
is 
.
See also
| 1989 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 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 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
