1992 AIME Problems/Problem 8
Problem
For any sequence of real numbers , define to be the sequence , whose $n^\mbox{th}_{}$ (Error compiling LaTeX. ! Missing { inserted.) term is . Suppose that all of the terms of the sequence are , and that . Find .
Solution
Note that the s are reminiscent of differentiation; from the condition , we are led to consider the differential equation This inspires us to guess a quadratic with leading coefficient 1/2 as the solution; as we must have roots at and .
Thus, .
Solution 2
Let , and .
Note that in every sequence of ,
Then
Since ,
Solving, .
Solution 3
Write out and add first terms of the second finite difference sequence:
…
…
…
Adding the above equations we get:
Now taking sum to in equation we get:
Now taking sum to in equation we get:
gives .
Kris17
See also
1992 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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