1984 AHSME Problems/Problem 12
Problem
If the sequence is defined by
where .
Then equals
Solution
We begin to evaluate the first couple of terms of the sequence, hoping to find a pattern: . We notice that the difference between succesive terms of the sequence are , a clear pattern. We can see that this pattern continues infinitely because of the recursive definition: each term is the previous term plus the next even number. Therefore, since the differences of consecutive terms form an arithmetic sequence, then the terms satisfy a quadratic, specifically, the one that contains the points , and . Let the quadratic be , so:
(1)
(2)
(3)
Subtracting (1) from (2) and (2) from (3) yields the two-variable system of equations
(4)
(5)
We can subtract (4) from (5) to find that , so . Substituting this back in yields , and substituting these back into one of the original equations yields , so the closed form for the terms is
, or
.
Substituting in yields .
Alternate Solution
Term can be viewed as the first term, , plus the arithmetic series .
By summing the numbers that form the arithmetic sequence with , we get .
See Also
1984 AHSME (Problems • Answer Key • Resources) | ||
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Followed by Problem 13 | |
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