2022 AMC 10A Problems/Problem 20
Problem
A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are ,
, and
. What is the fourth term of this sequence?
Solution
Let the arithmetic sequence be and the geometric sequence be
We are given that
and we wish to find
Subtracting the first equation from the second and the second equation from the third, we get
Subtract these results, we get
Note that or
We proceed with casework:
- If
then
and
The arithmetic sequence is
arriving at a contradiction.
- If
then
and
The arithmetic sequence is
and the geometric sequence is
This case is valid.
Therefore, The answer is
~mathboy282 ~MRENTHUSIASM
Video Solution by OmegaLearn
~ pi_is_3.14
See Also
2022 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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All AMC 10 Problems and Solutions |
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