determine the number of terms in the arithmetic sequence 57+ 63 + 69 +... +777

To determine the number of terms in the arithmetic sequence \( 57 + 63 + 69 + \dots + 777 \), we can follow these steps:

### Step 1: Identify the first term, common difference, and the last term.
- The first term \( a_1 = 57 \).
- The common difference \( d = 63 - 57 = 6 \).
- The last term \( a_n = 777 \).

### Step 2: Use the formula for the \( n \)-th term of an arithmetic sequence:
\[
a_n = a_1 + (n - 1) \cdot d
\]
where:
- \( a_n \) is the last term,
- \( a_1 \) is the first term,
- \( d \) is the common difference,
- \( n \) is the number of terms.

### Step 3: Plug in the known values and solve for \( n \):
\[
777 = 57 + (n - 1) \cdot 6
\]
First, subtract 57 from both sides:
\[
777 - 57 = (n - 1) \cdot 6
\]
\[
720 = (n - 1) \cdot 6
\]
Now, divide both sides by 6:
\[
120 = n - 1
\]
Finally, add 1 to both sides:
\[
n = 121
\]

### Conclusion:
The number of terms in the arithmetic sequence is \( \boxed{121} \).

Do you want further details or explanations on any step? Here are five related questions you might find interesting:
1. How do you derive the formula for the \( n \)-th term of an arithmetic sequence?
2. What is the sum of all the terms in this sequence?
3. How do arithmetic sequences differ from geometric sequences?
4. Can the number of terms in a sequence ever be a fraction?
5. How can you graphically represent an arithmetic sequence?

**Tip:** The sum of an arithmetic sequence can be calculated using the formula \( S_n = \frac{n}{2} \cdot (a_1 + a_n) \).

Determine the number of terms in the arithmetic sequence 57 + 63 + 69 + ... + 777.

Learn how to determine the number of terms in the arithmetic sequence starting from 57 and increasing by 6 up to 777. This problem involves applying the nth-term formula of an arithmetic sequence and is suitable for students in Grades 8-10.

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