Difference between revisions of "2003 AIME II Problems/Problem 8"
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== Solution == | == Solution == | ||
| − | If you multiply the corresponding terms of two arithmetic sequences, you get the terms of a quadratic function. Thus, we have a quadratic <math>ax^2+bx+c</math> such that f(1)=1440, f(2)=1716, and f(3)=1848. Plugging in the values for x gives us a system of three equations: | + | If you multiply the corresponding terms of two arithmetic sequences, you get the terms of a quadratic function. Thus, we have a quadratic <math>ax^2+bx+c</math> such that <math>f(1)=1440</math>, <math>f(2)=1716</math>, and <math>f(3)=1848</math>. Plugging in the values for x gives us a system of three equations: |
| − | |||
| − | 4a+2b+c=1716 | + | <math>a+b+c=1440</math> |
| − | + | ||
| − | 9a+3b+c=1848 | + | <math>4a+2b+c=1716</math> |
| + | |||
| + | <math>9a+3b+c=1848</math> | ||
Solving gives a=-72, b=492, and c=1020. Thus, the answer is <math>-72(8)^2+492\cdot8+1020=348</math> | Solving gives a=-72, b=492, and c=1020. Thus, the answer is <math>-72(8)^2+492\cdot8+1020=348</math> | ||
Revision as of 17:14, 21 January 2008
Problem
Find the eighth term of the sequence 
whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.
Solution
If you multiply the corresponding terms of two arithmetic sequences, you get the terms of a quadratic function. Thus, we have a quadratic 
such that 
, 
, and 
. Plugging in the values for x gives us a system of three equations:
Solving gives a=-72, b=492, and c=1020. Thus, the answer is 
See also
| 2003 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
