2006 AMC 12A Problems/Problem 23
Contents
[hide]Problem
Given a finite sequence of
real numbers, let
be the sequence
of
real numbers. Define
and, for each integer
,
, define
. Suppose
, and let
. If
, then what is
?
Solution 1
In general, such that
has
terms. Specifically,
To find x, we need only solve the equation
. Algebra yields
.
Solution 2
For every sequence of at least three terms,
$\[
A^2(S)=\left(\frac{a_1+2a_2+a_3}{4},\frac{a_2+2a_3+a_4}{4},\dots,\frac{a_{n-2}+2a_{n-1}+a_n}{4}\right).\]$ (Error compiling LaTeX. Unknown error_msg)Thus for , the coefficients of the terms in the numerator of
are the binomial coefficients
, and the denominator is
. Because
for all integers
, the coefficients of the terms in the numerators of
are
for
. The definition implies that the denominator of each term in
is
. For the given sequence, the sole term in
is\[ \frac{1}{2^{100}} \sum_{m=0}^{100} \binom{100}{m}a_{m+1} =
\frac{1}{2^{100}} \sum_{m=0}^{100} \binom{100}{m}x^m =
\frac{1}{2^{100}}(x+1)^{100}.\]Therefore\[
\left(\frac{1}{2^{50}}\right)=A^{100}(S)=\left(\frac{(1+x)^{100}}{2^{100}}\right),
\]so
, and because
, we have
.
- Alcumus
See also
2006 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
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 | |
All AMC 12 Problems and Solutions |
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