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

To find the sum of all the terms, including the seven arithmetic means between 3 and 35, we first need to determine the terms in the arithmetic sequence.

### Step 1: Determine the number of terms in the sequence
There are seven arithmetic means between 3 and 35. So, including 3 and 35, the total number of terms \(n\) in the sequence is:
\[
n = 7 + 2 = 9
\]

### Step 2: Determine the common difference \(d\)
The general formula for the \(n\)-th term of an arithmetic sequence is:
\[
a_n = a_1 + (n-1)d
\]
Here:
- \(a_1 = 3\) (the first term)
- \(a_9 = 35\) (the ninth term)

Substitute the known values into the formula for the ninth term:
\[
35 = 3 + (9-1)d
\]
\[
35 = 3 + 8d
\]
\[
8d = 35 - 3 = 32
\]
\[
d = \frac{32}{8} = 4
\]
So, the common difference \(d\) is 4.

### Step 3: Determine all the terms
Now that we have the common difference \(d = 4\), we can find all the terms of the sequence:
- First term \(a_1 = 3\)
- Second term \(a_2 = 3 + 4 = 7\)
- Third term \(a_3 = 7 + 4 = 11\)
- Fourth term \(a_4 = 11 + 4 = 15\)
- Fifth term \(a_5 = 15 + 4 = 19\)
- Sixth term \(a_6 = 19 + 4 = 23\)
- Seventh term \(a_7 = 23 + 4 = 27\)
- Eighth term \(a_8 = 27 + 4 = 31\)
- Ninth term \(a_9 = 35\)

So, the sequence is:
\[
3, 7, 11, 15, 19, 23, 27, 31, 35
\]

### Step 4: Calculate the sum of the sequence
The sum \(S_n\) of an arithmetic sequence can be calculated using the formula:
\[
S_n = \frac{n}{2} \times (a_1 + a_n)
\]
Substituting the known values:
\[
S_9 = \frac{9}{2} \times (3 + 35) = \frac{9}{2} \times 38 = \frac{9 \times 38}{2} = \frac{342}{2} = 171
\]

### Final Answer
The sum of all the terms in the sequence is **171**.

Would you like further details or have any questions about this? Here are five related questions you might find interesting:

1. How do you find the number of terms in an arithmetic sequence when only the first and last terms are given?
2. How can you determine the common difference if you only know the first term and the total sum of the sequence?
3. What is the formula for the nth term of an arithmetic sequence and how can it be used in different scenarios?
4. How would you find the sum of a sequence if the number of terms was unknown but the common difference was given?
5. How does the sum of an arithmetic series change if the common difference is negative?

**Tip:** When dealing with arithmetic sequences, always check if you have all the required values (first term, last term, number of terms, or common difference) to use the appropriate formulas efficiently.

Calculate the sum of arithmetic means between 3 and 35 using the arithmetic sequence formula. Understand how to find the number of terms, common difference, and apply the sum formula effectively. Suitable for Grades 7-9.

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