+0

# Arithmetic sequences

0
109
1

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

### 1+0 Answers

#1
+9461
0

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