Difference between revisions of "1984 AHSME Problems/Problem 12"
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== Alternate Solution == | == Alternate Solution == | ||
− | Term <math>a_{100}</math> can be viewed as the first term, <math>2</math>, plus the | + | Term <math>a_{100}</math> can be viewed as the first term, <math>2</math>, plus the arithmetic series <math> 2 + 4 +\cdots + 198</math>. |
− | By summing the | + | By summing the numbers that form the arithmetic sequence with <math>a_{1}</math>, we get <math>2 + \dfrac{2 + 198}{2}\cdot99 = 9902</math>. |
==See Also== | ==See Also== | ||
{{AHSME box|year=1984|num-b=12|num-a=13}} | {{AHSME box|year=1984|num-b=12|num-a=13}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 18:10, 30 November 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
.
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) | ||
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 |
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