Difference between revisions of "1992 AIME Problems/Problem 8"
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== Alternate Solution == | == Alternate Solution == | ||
+ | Let <math>\Delta(\Delta(...A)...)</math>, with <math>n\space\Delta</math>'s, be denoted as <math>\Delta^n</math>. | ||
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Note that in every sequence of <math>a_i</math>, <math>a_n=\binom{n-1}{1}\Delta a_n + \binom{n-1}{2}\Delta^2 a_n +\binom{n-1}{3}\Delta^3 a_n + ...</math> | Note that in every sequence of <math>a_i</math>, <math>a_n=\binom{n-1}{1}\Delta a_n + \binom{n-1}{2}\Delta^2 a_n +\binom{n-1}{3}\Delta^3 a_n + ...</math> | ||
Revision as of 21:26, 15 March 2009
Contents
[hide]Problem
For any sequence of real numbers , define to be the sequence , whose $n^\mbox{th}_{}$ (Error compiling LaTeX. Unknown error_msg) term is . Suppose that all of the terms of the sequence are , and that . Find .
Solution
Since the second differences are all and , can be expressed explicitly by the quadratic: .
Thus, .
Alternate Solution
Let , with 's, be denoted as .
Note that in every sequence of ,
Then
Since ,
Solving,
See also
1992 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |