Difference between revisions of "1992 AIME Problems/Problem 8"
m |
|||
Line 8: | Line 8: | ||
== Alternate Solution == | == Alternate Solution == | ||
− | Let <math>\Delta | + | Let <math>\Delta^1 A=\Delta A</math>, and <math>\Delta^n A=\Delta(\Delta^{(n-1)}A)</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> | 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> | ||
Line 20: | Line 20: | ||
<math>a_{92}=0=a_1+91(a_2-a_1)+\binom{91}{2}=91a_2-90a_1+4095</math> | <math>a_{92}=0=a_1+91(a_2-a_1)+\binom{91}{2}=91a_2-90a_1+4095</math> | ||
− | Solving, <math>a_1=\boxed{819}</math> | + | Solving, <math>a_1=\boxed{819}</math>. |
== See also == | == See also == | ||
{{AIME box|year=1992|num-b=7|num-a=9}} | {{AIME box|year=1992|num-b=7|num-a=9}} |
Revision as of 21:34, 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 , and .
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 |