To find the third arithmetic mean, we first need to understand that inserting five arithmetic means between $-9$ and $9$ creates a sequence with a total of 7 terms. The first term is $-9$ and the seventh term is $9$.

The general term for an arithmetic sequence can be written as:

$a_n = a_1 + (n-1)d$

Where:

• $a_n$ is the $n$-th term,
• $a_1$ is the first term (here, $-9$),
• $d$ is the common difference,
• $n$ is the term number.

Step 1: Calculate the common difference $d$

We know the first term ($a_1 = -9$) and the seventh term ($a_7 = 9$):

$a_7 = a_1 + (7-1)d$ $9 = -9 + 6d$ $9 + 9 = 6d$ $18 = 6d$ $d = \frac{18}{6} = 3$

So, the common difference is $d = 3$.

Step 2: Find the third arithmetic mean

The arithmetic means are the terms between the first and last terms. The third mean is the third term in the sequence, i.e., $a_3$.

Using the formula for the general term:

$a_3 = a_1 + (3-1)d$ $a_3 = -9 + 2 \times 3$ $a_3 = -9 + 6 = -3$

Thus, the third arithmetic mean is $-3$.

Would you like further details or have any questions?

Here are 5 related questions to deepen understanding:

1. What is the sum of all the terms in the arithmetic sequence?
2. How would the means change if we inserted seven arithmetic means instead?
3. What is the formula for the sum of an arithmetic series, and how can it be applied here?
4. How do you find the median of an arithmetic sequence?
5. How would you insert geometric means instead of arithmetic means between $-9$ and $9$?

Tip: The common difference in an arithmetic sequence is the key to understanding how each term is spaced. Always solve for it first!