Difference between revisions of "2022 AMC 12B Problems/Problem 9"
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== Problem == | == Problem == | ||
− | The sequence <math>a_0,a_1,a_2,\cdots</math> is a strictly increasing arithmetic sequence of positive integers such that<cmath>2^{a_7}=2^{27} \cdot a_7.</cmath>What is the minimum possible value of <math>a_2</math>? | + | The sequence <math>a_0,a_1,a_2,\cdots</math> is a strictly increasing arithmetic sequence of positive integers such that <cmath>2^{a_7}=2^{27} \cdot a_7.</cmath> What is the minimum possible value of <math>a_2</math>? |
<math>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 16 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 22</math> | <math>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 16 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 22</math> |
Revision as of 21:19, 17 November 2022
Problem
The sequence is a strictly increasing arithmetic sequence of positive integers such that
What is the minimum possible value of
?
Solution 1
We can rewrite the given equation as . Hence,
must be a power of
and larger than
. The first power of 2 that is larger than
, namely
, does satisfy the equation:
. In fact, this is the only solution because
is exponential whereas
is linear, so their graphs will not intersect again.
Now, let the common difference in the sequence be . Hence,
and
. To minimize
, we maxmimize
. Since the sequence contains only positive integers,
and hence
. When
,
.
See also
2022 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.