Given the first term and the common differences of an arithmetic sequence find the recursive formula and three terms in the sequence after the last one given a one= 3/5, d=-1/3

To find the recursive formula for the arithmetic sequence and the three terms after the last one, we follow these steps:

### Step 1: Formula for an Arithmetic Sequence
The recursive formula for an arithmetic sequence is:
\[
a_n = a_{n-1} + d
\]
where:
- \( a_n \) is the \(n\)-th term,
- \( a_{n-1} \) is the previous term,
- \( d \) is the common difference.

### Step 2: Given Information
- First term (\( a_1 \)) = \( \frac{3}{5} \),
- Common difference (\( d \)) = \( -\frac{1}{3} \).

Thus, the recursive formula for the given sequence is:
\[
a_n = a_{n-1} - \frac{1}{3}
\]

### Step 3: Find the Next Three Terms
Using the recursive formula, we can calculate the next terms after \( a_1 \):

- **Second term**: 
\[
a_2 = a_1 - \frac{1}{3} = \frac{3}{5} - \frac{1}{3}
\]
To subtract these fractions, we find a common denominator:
\[
a_2 = \frac{9}{15} - \frac{5}{15} = \frac{4}{15}
\]

- **Third term**: 
\[
a_3 = a_2 - \frac{1}{3} = \frac{4}{15} - \frac{5}{15} = -\frac{1}{15}
\]

- **Fourth term**: 
\[
a_4 = a_3 - \frac{1}{3} = -\frac{1}{15} - \frac{5}{15} = -\frac{6}{15} = -\frac{2}{5}
\]

### Recursive Formula:
\[
a_n = a_{n-1} - \frac{1}{3}
\]

### Next Three Terms:
1. \( a_2 = \frac{4}{15} \)
2. \( a_3 = -\frac{1}{15} \)
3. \( a_4 = -\frac{2}{5} \)

Would you like any further explanation or details? Here are 5 follow-up questions to consider:

1. What is the explicit formula for this arithmetic sequence?
2. How do you determine the general term of an arithmetic sequence?
3. How would you find the sum of the first \( n \) terms of this sequence?
4. How does the common difference affect the sequence's behavior over time?
5. What are some real-life examples of arithmetic sequences?

**Tip**: Always ensure to convert fractions to a common denominator when adding or subtracting them.

Given the first term and the common differences of an arithmetic sequence, find the recursive formula and three terms in the sequence after the last one. Given a1 = 3/5, d = -1/3.

Learn how to find the recursive formula and calculate the next three terms of an arithmetic sequence with the first term 3/5 and common difference -1/3. This solution involves recursive sequences and fraction operations, suitable for students in Grades 8-10.

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