how do you find the sum of the first 15 terms in an arithmetic sequence? the first term is 7 and the common difference is 4

To find the sum of the first 15 terms in an arithmetic sequence where the first term \( a_1 \) is 7 and the common difference \( d \) is 4, you can use the formula for the sum of the first \( n \) terms of an arithmetic sequence:

\[ S_n = \frac{n}{2} \left( 2a_1 + (n-1)d \right) \]

Here’s how you can apply this formula step-by-step:

1. **Identify the values:**
   - \( a_1 = 7 \) (first term)
   - \( d = 4 \) (common difference)
   - \( n = 15 \) (number of terms)

2. **Plug the values into the formula:**
   \[
   S_{15} = \frac{15}{2} \left( 2 \cdot 7 + (15-1) \cdot 4 \right)
   \]

3. **Simplify inside the parentheses:**
   \[
   S_{15} = \frac{15}{2} \left( 14 + 14 \cdot 4 \right)
   \]

4. **Calculate the multiplication and addition:**
   \[
   S_{15} = \frac{15}{2} \left( 14 + 56 \right)
   \]

5. **Combine the terms:**
   \[
   S_{15} = \frac{15}{2} \cdot 70
   \]

6. **Multiply and simplify:**
   \[
   S_{15} = \frac{1050}{2} = 525
   \]

So, the sum of the first 15 terms of the arithmetic sequence is \( 525 \).

Would you like more details or have any questions?

Here are 8 relative questions for further practice:
1. Find the sum of the first 20 terms of an arithmetic sequence where the first term is 5 and the common difference is 3.
2. If the first term is 8 and the common difference is 6, what is the sum of the first 10 terms?
3. Calculate the sum of the first 12 terms in an arithmetic sequence with the first term 10 and the common difference 2.
4. What is the sum of the first 25 terms of an arithmetic sequence if the first term is 1 and the common difference is 5?
5. Find the sum of the first 30 terms of an arithmetic sequence where the first term is 4 and the common difference is 3.
6. Determine the sum of the first 50 terms of an arithmetic sequence with the first term 2 and the common difference 4.
7. If the first term is -3 and the common difference is 7, what is the sum of the first 15 terms?
8. Calculate the sum of the first 18 terms in an arithmetic sequence where the first term is 9 and the common difference is 5.

**Tip**: Remember that the sum of the first \( n \) terms of an arithmetic sequence can also be found using the alternative formula \( S_n = \frac{n}{2} (a_1 + a_n) \) where \( a_n \) is the nth term. This is particularly useful if you already know the last term.

Learn how to calculate the sum of the first 15 terms in an arithmetic sequence where the first term is 7 and the common difference is 4. This problem utilizes the formula for the sum of an arithmetic sequence, providing a step-by-step solution.

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