Difference between revisions of "2002 AMC 12B Problems/Problem 22"
(→Solution 3) |
(→Solution 2) |
||
Line 22: | Line 22: | ||
~yofro | ~yofro | ||
+ | |||
+ | == Solution 3 == | ||
+ | |||
+ | Note that <math>a_2 = \frac{1}{\log_2 2022}</math>. <math>1</math> is also equal to <math>\log_2 2</math>. So <math>a_2 = \frac{\log_2 2}{\log_2 2022}</math>. By the change of bases formula, <math>a_2 = \log_{2022} 2</math>. Following the same reasoning, <math>a_3 = \log_{2022} 3</math>, <math>a_4 = \log_{2022} 4</math> and so on. | ||
+ | |||
+ | <cmath>b = \log_{2022} 2 + \log_{2022} 3 + .....+ \log_{2022} 5 = \log_{2022} 5! = \log_{2022} 120</cmath> | ||
+ | Now solving for <math>c</math>, we see that it equals <math>\log_{2022} (10\cdot 11 \cdot 12 \cdot 13 \cdot 14)</math> | ||
+ | <cmath>b-c = \log_{2022} 120 - \log_{2022} 240240 \rightarrow \log_{2022} \frac{1}{2022} = \boxed{-1}</cmath> | ||
+ | |||
+ | ~YBSuburbanTea | ||
== See also == | == See also == |
Revision as of 18:21, 13 January 2022
Problem
For all integers greater than
, define
. Let
and
. Then
equals
Solution
By the change of base formula, . Thus
Solution 2
Note that . Thus
. Also notice that if we have a log sum, we multiply, and if we have a log product, we divide. Using these properties, we get that the result is the following:
~yofro
Solution 3
Note that .
is also equal to
. So
. By the change of bases formula,
. Following the same reasoning,
,
and so on.
Now solving for
, we see that it equals
~YBSuburbanTea
See also
2002 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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 AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.