Insert three terms in the 2 means 14 to form a series?

To find three terms to insert between 2 and 14 that form an arithmetic series, we need to calculate the common difference and then the terms themselves.

### Step 1: Define the Arithmetic Sequence
Let the sequence be \(2, a_1, a_2, a_3, 14\), where \(a_1\), \(a_2\), and \(a_3\) are the three terms to insert.

### Step 2: Calculate the Common Difference
The sequence has 5 terms, so the common difference \(d\) can be calculated as follows:
\[
d = \frac{\text{Last term} - \text{First term}}{n + 1} = \frac{14 - 2}{4} = \frac{12}{4} = 3
\]

### Step 3: Find the Terms
Now, using the common difference \(d = 3\):
- \(a_1 = 2 + 3 = 5\)
- \(a_2 = 5 + 3 = 8\)
- \(a_3 = 8 + 3 = 11\)

### Final Sequence
The complete arithmetic series is \(2, 5, 8, 11, 14\).

Would you like more details or have any questions?

Here are five related questions you might find interesting:
1. How do you determine the common difference in an arithmetic series?
2. What happens if you want to insert more or fewer terms between two numbers?
3. How would the sequence change if it were a geometric progression instead of arithmetic?
4. Can you derive a general formula for inserting \(n\) terms between two given numbers?
5. How do arithmetic sequences apply in real-world scenarios, such as finance or physics?

**Tip:** When working with sequences, always start by identifying the number of terms and whether the sequence is arithmetic or geometric.

Learn how to insert three terms in an arithmetic sequence between 2 and 14 with a detailed step-by-step solution. This problem covers arithmetic sequences, common differences, and term calculation in sequences. Suitable for students in Grades 7-9.

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