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2004 Mathcounts State Sprint #19: The
points (x, y) represented in this table lie on a straight line. The
point (13, q) lies on the same line. What is the value of p + q? Express
your answer as a decimal to the nearest tenth.
Solution:
#19: Look at the table and you'll see each time x + 2, y would -3.
-5 to -14 is (-9), three times (-3) so p = 2 + 3 x 2 = 8
p would = 13 when 2 + 5.5 * 2 = 13 so q = -5 + (5.5) * (-3) = -21.5
p + q = -13.5
2004 Mathcounts State Sprint #24:
The terms x, x + 2, x + 4, ..., x + 2n form an arithmetic sequence,
with x an integer. If each term of the sequence is cubed, the sum of the
cubes is - 1197. What is the value of n if n > 3?
Solution:
The common difference in that arithmetic sequence is 2 and the sum of
the cubes is -1197, which means that these numbers are all odd numbers. (cubes
of odd numbers are odd and the sum of odd terms of odd numbers is odd. )
(-5)3 + (-7)3 + (-9)3 = -1197 However, n is larger than 3 (given) so the sequence will look like this:
(-9)3+ (-7)3+ (-5)3+ (-3)3+ (-1)3 + (1)3 + (3)3 = -1197
x = - 9 and x + 2n = 3, plug in and you get -9 + 2n = 3; n = 6
... nice
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