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# Urgent Help needed!

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Let $a_1, a_2, a_3,\dots$ be an arithmetic sequence. If $a_1 + a_3 + a_5 = -12$ and $a_1a_3a_5 = 80$, find all possible values of $a_{10}$.

I am unsure, is this some type of system of equations question?

May 17, 2022

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Yes, it is.

Let the starting term be $$a$$ and let the common difference be $$d$$.

The $$n_{\text {th}}$$ term of this sequence is $$a + d(n-1)$$. This means that the starting (1st) term is $$a$$, and the second term is $$a + d$$

Thus, we can form the following system: $$a+(a+2d)+(a+4d) = -12$$ and $$a(a+2d)(a+4d) = 80$$

Now, we just have to solve for the 10th term, or $$a + 9d$$.

Can you take it from here?

May 17, 2022