1992 AIME Problems/Problem 8
Problem
For any sequence of real numbers , define
to be the sequence
, whose
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
The sequence is the second finite difference sequence, and the first
terms of this sequence can be computed in terms of the original sequence as shown below.
Adding the above equations we find that
We can sum equation from
to
, finding
We can also sum equation from
to
, finding
Finally,
gives
.
Kris17
See also
1992 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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