2022 AMC 12B Problems/Problem 9
Contents
[hide]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;
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
,
.
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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 |
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