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AIME #8 2003 II
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AIME #8 2003 II
by stupidityismygam » Wed Jul 11, 2007 12:19 pm
Find the eighth term of the sequence [unparseable or potentially dangerous latex formula] whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.
Last edited by stupidityismygam on Wed Jul 11, 2007 1:45 pm, edited 1 time in total.
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by Roel » Wed Jul 11, 2007 1:07 pm
Lets say our first arithmetic sequence starts with the number [unparseable or potentially dangerous latex formula], and also has the common difference [unparseable or potentially dangerous latex formula]. Now our second sequence will start with the number [unparseable or potentially dangerous latex formula] with a common difference of [unparseable or potentially dangerous latex formula].
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
The eighth term:
[unparseable or potentially dangerous latex formula]
Substitute:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
The eighth term:
[unparseable or potentially dangerous latex formula]
Substitute:
[unparseable or potentially dangerous latex formula]
Last edited by Roel on Wed Jul 11, 2007 4:58 pm, edited 1 time in total.
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Re: AIME #8 2003 II
by Quelloquialism » Wed Jul 11, 2007 1:57 pm
My solution:
Define the first term of the first sequence to be [unparseable or potentially dangerous latex formula], the common difference of the first sequence to be [unparseable or potentially dangerous latex formula], the first term of the second sequence to be [unparseable or potentially dangerous latex formula], and the common difference of the second sequence to be [unparseable or potentially dangerous latex formula].
Equation 1: [unparseable or potentially dangerous latex formula]
Equation 2: [unparseable or potentially dangerous latex formula]
Equation 3: [unparseable or potentially dangerous latex formula]
Multiply out Equation 3.
Equation 4: [unparseable or potentially dangerous latex formula]
Multiply out Equation 2.
[unparseable or potentially dangerous latex formula]
Multiply the result by 2.
Equation 5: [unparseable or potentially dangerous latex formula]
Subtract Equation 4 from Equation 5.
[unparseable or potentially dangerous latex formula]
Substitute in Equation 1.
[unparseable or potentially dangerous latex formula]
Combine the constant values.
[unparseable or potentially dangerous latex formula]
Divide by -2.
Equation 6: [unparseable or potentially dangerous latex formula]
Now, take Equation 4 again.
[unparseable or potentially dangerous latex formula]
Substitute in Equation 1 and Equation 6.
[unparseable or potentially dangerous latex formula]
Combine the constant values.
Equation 7: [unparseable or potentially dangerous latex formula]
The eighth term should be...
[unparseable or potentially dangerous latex formula]
Multiply this out.
[unparseable or potentially dangerous latex formula]
Factor out the 7 in the middle two terms.
[unparseable or potentially dangerous latex formula]
Substitute in Equation 1, Equation 6, and Equation 7.
[unparseable or potentially dangerous latex formula]
Essentially the same solution.
Define the first term of the first sequence to be [unparseable or potentially dangerous latex formula], the common difference of the first sequence to be [unparseable or potentially dangerous latex formula], the first term of the second sequence to be [unparseable or potentially dangerous latex formula], and the common difference of the second sequence to be [unparseable or potentially dangerous latex formula].
Equation 1: [unparseable or potentially dangerous latex formula]
Equation 2: [unparseable or potentially dangerous latex formula]
Equation 3: [unparseable or potentially dangerous latex formula]
Multiply out Equation 3.
Equation 4: [unparseable or potentially dangerous latex formula]
Multiply out Equation 2.
[unparseable or potentially dangerous latex formula]
Multiply the result by 2.
Equation 5: [unparseable or potentially dangerous latex formula]
Subtract Equation 4 from Equation 5.
[unparseable or potentially dangerous latex formula]
Substitute in Equation 1.
[unparseable or potentially dangerous latex formula]
Combine the constant values.
[unparseable or potentially dangerous latex formula]
Divide by -2.
Equation 6: [unparseable or potentially dangerous latex formula]
Now, take Equation 4 again.
[unparseable or potentially dangerous latex formula]
Substitute in Equation 1 and Equation 6.
[unparseable or potentially dangerous latex formula]
Combine the constant values.
Equation 7: [unparseable or potentially dangerous latex formula]
The eighth term should be...
[unparseable or potentially dangerous latex formula]
Multiply this out.
[unparseable or potentially dangerous latex formula]
Factor out the 7 in the middle two terms.
[unparseable or potentially dangerous latex formula]
Substitute in Equation 1, Equation 6, and Equation 7.
[unparseable or potentially dangerous latex formula]
Essentially the same solution.
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