An arithmetic sequence has S1=-7 and S3=0 . Find S30

To find \( S_{30} \) in an arithmetic sequence, we need to determine the common difference \( d \) and then use the formula for the \( n \)-th term of an arithmetic sequence.

### Step 1: Identify the known terms
- First term \( S_1 = -7 \)
- Third term \( S_3 = 0 \)

### Step 2: Use the formula for the \( n \)-th term
The formula for the \( n \)-th term of an arithmetic sequence is:
\[
S_n = S_1 + (n-1) \cdot d
\]
For \( S_3 \), we have:
\[
S_3 = S_1 + 2d
\]
Substitute the known values:
\[
0 = -7 + 2d
\]

### Step 3: Solve for the common difference \( d \)
\[
2d = 7 \quad \Rightarrow \quad d = \frac{7}{2} = 3.5
\]

### Step 4: Find \( S_{30} \)
Using the formula again:
\[
S_{30} = S_1 + 29d
\]
Substitute the values of \( S_1 \) and \( d \):
\[
S_{30} = -7 + 29 \times 3.5
\]
\[
S_{30} = -7 + 101.5 = 94.5
\]

Thus, \( S_{30} = 94.5 \).

Would you like more details or have any questions?

Here are 5 related questions you might explore:
1. How do you find the common difference if you only know the first and last terms?
2. What is the formula for the sum of the first \( n \) terms in an arithmetic sequence?
3. How can you determine the number of terms in an arithmetic sequence if you know the first and last terms?
4. What happens to an arithmetic sequence if the common difference is negative?
5. How do you find a specific term if you're given two non-consecutive terms?

**Tip:** In arithmetic sequences, knowing just two terms and their positions is sufficient to determine the entire sequence.

An arithmetic sequence has S1=-7 and S3=0. Find S30.

Learn how to find the 30th term (S30) of an arithmetic sequence where S1=-7 and S3=0. This step-by-step solution covers the arithmetic sequence formula and finding the common difference (d). Ideal for students in Grades 9-10.

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