There are seven arithmetic means between 3 and 35. Find the sum of all terms.

We are given that there are seven arithmetic means between 3 and 35. This means that 3 and 35 are the first and the ninth terms of an arithmetic sequence.

In an arithmetic sequence, the sum of terms can be found using the formula for the sum of an arithmetic sequence:

\[
S_n = \frac{n}{2} \times (a + l)
\]

Where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( n \) is the number of terms,
- \( a \) is the first term,
- \( l \) is the last term.

Here:
- \( a = 3 \) (the first term),
- \( l = 35 \) (the last term),
- \( n = 9 \) (since there are seven arithmetic means between 3 and 35, making a total of nine terms).

Now, substitute these values into the sum formula:

\[
S_9 = \frac{9}{2} \times (3 + 35) = \frac{9}{2} \times 38 = 9 \times 19 = 171
\]

So, the sum of all the terms is \( 171 \).

Would you like more details or have any questions?

Here are five related questions to extend your understanding:
1. How do you find the common difference in an arithmetic sequence?
2. What is the formula for the nth term of an arithmetic sequence?
3. How can we derive the number of arithmetic means between two numbers?
4. What is the sum of the first 20 terms of an arithmetic sequence where the first term is 5 and the common difference is 3?
5. Can you determine the general term of an arithmetic sequence if the first term is 7 and the common difference is 4?

**Tip:** The common difference in an arithmetic sequence can be found by subtracting consecutive terms, or by using the formula \( d = \frac{l - a}{n-1} \).

Learn how to find the sum of an arithmetic sequence that includes seven arithmetic means between 3 and 35. The solution involves key concepts of arithmetic sequences, including the sum formula and calculating the total of nine terms. Suitable for students in Grades 8-10.

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