Difference between revisions of "Mock AIME 3 Pre 2005 Problems/Problem 6"
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which obtains equation <math>C</math>. | which obtains equation <math>C</math>. | ||
− | + | Adding equations <math>A</math> and <math>C</math> yields | |
<cmath>2a^2=n+1</cmath> | <cmath>2a^2=n+1</cmath> | ||
<cmath>a=\sqrt{\frac{n+1}{2}}</cmath> | <cmath>a=\sqrt{\frac{n+1}{2}}</cmath> |
Revision as of 14:09, 28 November 2019
Contents
[hide]Problem
Let denote the value of the sum
can be expressed as
, where
and
are positive integers and
is not divisible by the square of any prime. Determine
.
Solution
Notice that . Thus, we have
This is a telescoping series; note that when we expand the summation, all of the intermediary terms cancel, leaving us with
, and
.
Solution 2
Simplifying the expression yields
Now we can assume that
for some
,
,
.
Squaring the first equation yields
which gives the system of equations
calling them equations
and
, respectively.
Also we have
which obtains equation
.
Adding equations and
yields
See Also
Mock AIME 3 Pre 2005 (Problems, Source) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |