Difference between revisions of "1984 AHSME Problems/Problem 12"
(Created page with "==Problem== If the sequence <math> \{a_n\} </math> is defined by <math> a_1=2 </math> <math> a_{n+1}=a_n+2n </math> where <math> n\geq1 </math>. Then <math> a_{100} </mat...") |
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Revision as of 11:49, 5 July 2013
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 
.
See Also
| 1984 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 13 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
