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Write the formula for the nth term of the given arithmetic sequence

The image shows two arithmetic sequences:

1. Sequence 1: \( 1, 4, 7, 10, 13, \dots \)
2. Sequence 2: \( \frac{23}{5}, \frac{48}{5}, \frac{73}{5}, \frac{98}{5}, 123, \dots \)

### Finding the nth term for both sequences:

#### 1. Sequence 1: \( 1, 4, 7, 10, 13, \dots \)

This is an arithmetic sequence where the first term \( a_1 = 1 \) and the common difference \( d = 3 \). The general formula for the \( n \)-th term of an arithmetic sequence is:

\[
a_n = a_1 + (n - 1) d
\]

Substituting the values:
\[
a_n = 1 + (n - 1) \cdot 3 = 1 + 3n - 3 = 3n - 2
\]

So, the nth term of the first sequence is:
\[
a_n = 3n - 2
\]

#### 2. Sequence 2: \( \frac{23}{5}, \frac{48}{5}, \frac{73}{5}, \frac{98}{5}, \dots \)

This is also an arithmetic sequence, where the first term \( a_1 = \frac{23}{5} \) and the common difference is:
\[
d = \frac{48}{5} - \frac{23}{5} = \frac{25}{5} = 5
\]

Using the general formula again:
\[
a_n = a_1 + (n - 1) d
\]

Substituting the values:
\[
a_n = \frac{23}{5} + (n - 1) \cdot 5 = \frac{23}{5} + 5n - 5
\]

Simplifying:
\[
a_n = 5n + \frac{23}{5} - \frac{25}{5} = 5n - \frac{2}{5}
\]

So, the nth term of the second sequence is:
\[
a_n = 5n - \frac{2}{5}
\]

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Would you like further details or have any questions?

Here are five related questions:
1. How do we determine the common difference in an arithmetic sequence?
2. What is the relationship between the first term and the common difference in a sequence?
3. How can you find the sum of the first \( n \) terms of an arithmetic sequence?
4. What is the nth term formula for a geometric sequence?
5. How can you extend an arithmetic sequence beyond the 10th term?

**Tip:** To identify an arithmetic sequence, look for a constant difference between consecutive terms.

Write the formula for the nth term of the given arithmetic sequence.

Learn how to find the nth term of arithmetic sequences such as 1, 4, 7, 10, ... and 23/5, 48/5, 73/5, 98/5, ... This problem covers key algebraic concepts including sequence formulas and arithmetic progressions. Suitable for students in Grades 8-10.

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