Insert 3 terms in the 2 means 14 to form a series

To insert three terms between 2 and 14 to form an arithmetic series, we can follow these steps:

1. **Identify the terms of the sequence:**
   Let's denote the three terms we need to find as \(a_1\), \(a_2\), and \(a_3\). So the sequence would be: \(2, a_1, a_2, a_3, 14\).

2. **Arithmetic sequence formula:**
   In an arithmetic sequence, the difference between consecutive terms is constant. If we let \(d\) be the common difference, then the terms of the sequence can be written as:
   - First term: \(2\) (given)
   - Second term: \(2 + d = a_1\)
   - Third term: \(2 + 2d = a_2\)
   - Fourth term: \(2 + 3d = a_3\)
   - Fifth term: \(2 + 4d = 14\) (given)

3. **Set up the equation:**
   From the fifth term, we have:
   \[
   2 + 4d = 14
   \]

4. **Solve for \(d\):**
   \[
   4d = 14 - 2
   \]
   \[
   4d = 12
   \]
   \[
   d = 3
   \]

5. **Find the inserted terms:**
   Using \(d = 3\), we can now find \(a_1\), \(a_2\), and \(a_3\):
   - \(a_1 = 2 + d = 2 + 3 = 5\)
   - \(a_2 = 2 + 2d = 2 + 2 \times 3 = 8\)
   - \(a_3 = 2 + 3d = 2 + 3 \times 3 = 11\)

Therefore, the sequence is: **2, 5, 8, 11, 14**.

This sequence is arithmetic, with a common difference of 3.

### Do you want details or have any questions?

### Related Questions:

1. What is a geometric series, and how is it different from an arithmetic series?
2. How do you find the common difference if more terms of the sequence are provided?
3. What happens if more than three terms need to be inserted between two numbers?
4. Can the common difference in an arithmetic sequence be a fraction or a negative number?
5. How do you determine the sum of an arithmetic series?

### Tip:

When inserting terms to form an arithmetic sequence, always identify the total number of terms first, including the given terms, and then solve for the common difference \(d\).

Learn how to insert three terms between 2 and 14 to form an arithmetic series. This problem involves identifying terms in an arithmetic sequence and calculating the common difference. Suitable for students in Grades 7-9.

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