how many terms are in the sequence 4,7,10,...61?

To determine the number of terms in the sequence \(4, 7, 10, \ldots, 61\), we can recognize this as an arithmetic sequence where:

- The first term \( a = 4 \).
- The common difference \( d = 7 - 4 = 3 \).

The general formula for the \( n \)-th term of an arithmetic sequence is:

\[
a_n = a + (n - 1) \cdot d
\]

We need to find \( n \) when the last term \( a_n = 61 \). Plugging in the values:

\[
61 = 4 + (n - 1) \cdot 3
\]

Simplify the equation:

\[
61 - 4 = (n - 1) \cdot 3
\]

\[
57 = (n - 1) \cdot 3
\]

Now, divide both sides by 3:

\[
n - 1 = \frac{57}{3}
\]

\[
n - 1 = 19
\]

So, 

\[
n = 20
\]

Therefore, there are **20 terms** in the sequence.

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Would you like a detailed explanation of any step? Here are some related questions:

1. How do you find the common difference in an arithmetic sequence?
2. What is the formula for the sum of an arithmetic sequence?
3. How do you find the \( n \)-th term of an arithmetic sequence?
4. Can this method be used for sequences with different patterns?
5. What changes if the sequence is geometric instead of arithmetic?

**Tip:** Always identify the common difference or ratio when dealing with sequences—it’s the key to finding missing terms or sums!

Learn how to find the number of terms in the arithmetic sequence 4, 7, 10, ..., 61. This step-by-step solution explains key concepts in sequences and series, including the nth term formula. Suitable for students in Grades 8-10.

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