Difference between revisions of "1969 Canadian MO Problems/Problem 6"
m (→Solution 1) |
m (→Solution 1) |
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<cmath>2!-1!+3!-2!+\cdots+n!-(n-1)!+(n+1)!-n!</cmath> | <cmath>2!-1!+3!-2!+\cdots+n!-(n-1)!+(n+1)!-n!</cmath> | ||
Which telescopes to | Which telescopes to | ||
− | <cmath>(n+1)!-1!= | + | <cmath>(n+1)!-1!=(n+1)!-1</cmath> |
+ | So <math>(n+1)!-1</math> is the solution. | ||
{{Old CanadaMO box|num-b=5|num-a=7|year=1969}} | {{Old CanadaMO box|num-b=5|num-a=7|year=1969}} |
Revision as of 09:00, 3 December 2015
Problem
Find the sum of , where
.
Solution 1
Note that for any positive integer
Hence, pairing terms in the series will telescope most of the terms.
If is odd,
If is even,
In both cases, the expression telescopes into
Solution 1
We need to evaluate
We replace
with
Distribution yields
Simplifying,
Which telescopes to
So
is the solution.
1969 Canadian MO (Problems) | ||
Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Problem 7 |