Difference between revisions of "Mock AIME 1 2006-2007 Problems/Problem 9"
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− | {{ | + | Let the [[ratio]] of consecutive terms of the sequence be <math>r \in \mathbb{C}</math>. Then we have by the given that <math>1 = a_{10} = r^{10} a_0 = 1024r^{10}</math> so <math>r^{10} = 2^{-10}</math> and <math>r = \frac \omega 2</math>, where <math>\omega</math> can be any of the tenth [[roots of unity]]. |
− | --- | + | Then the sum <math>S = a_{10} + a_{11} + \ldots = 1 + r + r^2 +\ldots = \frac{1}{1-r}</math> has value <math>\frac 1{1 - \omega / 2}</math>. Different choices of <math>\omega</math> clearly lead to different values for <math>S</math>, so we don't need to worry about the distinctness condition in the problem. Then the value we want is <math>\sum_{\omega^{10} = 1} \sum_{i = 10}^\infty 1024 \left(\frac\omega2\right)^i = 1024 \sum_{i = 10}^\infty 2^{-i} \sum_{\omega^{10}=1} \omega^i</math>. Now, recall that if <math>z_1, z_2, \ldots, z_n</math> are the <math>n</math> <math>n</math>th [[root of unity | roots of unity]] then for any [[integer]] <math>m</math>, <math>z_1^m + \ldots + z_n^m</math> is 0 unless <math>n | m</math> in which case it is 1. Thus this simplifies to ... |
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+ | ==See Also== | ||
*[[Mock AIME 1 2006-2007/Problem 8 | Previous Problem]] | *[[Mock AIME 1 2006-2007/Problem 8 | Previous Problem]] |
Revision as of 23:10, 12 November 2006
Problem
Revised statement
Let be a geometric sequence of complex numbers with
and
, and let
denote the infinite sum
. If the sum of all possible distinct values of
is
where
and
are relatively prime positive integers, compute the sum of the positive prime factors of
.
Original statement
Let be a geometric sequence for
with
and
. Let
denote the infinite sum:
. If the sum of all distinct values of
is
where
and
are relatively prime positive integers, then compute the sum of the positive prime factors of
.
Solution
Let the ratio of consecutive terms of the sequence be . Then we have by the given that
so
and
, where
can be any of the tenth roots of unity.
Then the sum has value
. Different choices of
clearly lead to different values for
, so we don't need to worry about the distinctness condition in the problem. Then the value we want is
. Now, recall that if
are the
th roots of unity then for any integer
,
is 0 unless
in which case it is 1. Thus this simplifies to ...