2013 AMC 12A Problems/Problem 14
Problem
The sequence
,
,
,
,
is an arithmetic progression. What is ?
Solution
Since the sequence is arithmetic,
+
=
, where
is the common difference.
Therefore,
=
-
=
, and
=
(
) =
Now that we found , we just add it to the first term to find
:
+
=
=
=
=
=
, which is
Alternate solution
As the sequence ,
,
,
,
is an arithmetic progression, the sequence
must be a geometric progression.
If we factor the two known terms we get and
, thus the quotient is obviously
and therefore
.
See also
2013 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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.