2022 AMC 10A Problems/Problem 20
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?
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 either 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
Solution 2 (Elimination)
Start similarly to Solution 1 and deduce the three equations Then, add the last two equations and take away the first equation to get We can solve for this in terms of what we want: .
We're looking for . We can substitute our value of in here to get Since our sequence only has positive integers we can now check by the answer choices. For each answer choice, we can subtract and factor it to see if it has a perfect square factor and at least one other factor and those should differ by . From this, the only possible answer choices are and , where . To solve for , we look back to the given equations above.
We are looking for . If were the answer, then we know that would have to be divisible by and would equal . Looking at our second equation, if this were the case, then would also have to be divisible by . However, this contradicts the third equation, as all variables are divisible by , but their sum isn't. So, is our answer.
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