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Let a_1, a_2, a_3, be an arithmetic sequence. If a_1 + a_3 + a_5 = -12 and a_1a_3a_5 = \(120\), find all possible values of a_{10}.

 May 13, 2022
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I assume a_1, a_2, a_3, ... is an arithmetic sequence, because there are no a_5 in the problem.

 

For an arithmetic sequence a_n, \(a_{n - k} + a_{n + k} = 2a_n\). So \(a_1 + a_5 = 2a_3\) by plugging in n = 3, k = 2.

Therefore, 

 

\(3a_3 = -12\\ a_3 = -4\)

 

Now you just solve \(\begin{cases}a_1 + a_5 = -8\\a_1a_5 = -30\end{cases}\) to find the values of a_1 and a_5. That way, you will know the possible common differences of the arithmetic sequence. Then you can use \(a_{10} = a_1 + 9(\text{common difference})\) to calculate the possible values of a_{10}. 

 May 14, 2022

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