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To solve for the first term (\(a_1\)) and the common difference (\(d\)) of the arithmetic sequence, let's break it down:

### Information given:
- The 17th term of the sequence, \(a_{17} = -117\).
- The sum of the first 31 terms, \(S_{31} = -3317\).

### Step 1: Use the formula for the nth term of an arithmetic sequence
The general formula for the \(n\)-th term of an arithmetic sequence is:

\[
a_n = a_1 + (n - 1)d
\]

For the 17th term, we have:

\[
a_{17} = a_1 + (17 - 1)d
\]

Substitute the given value for \(a_{17}\):

\[
-117 = a_1 + 16d \tag{1}
\]

### Step 2: Use the formula for the sum of the first \(n\) terms of an arithmetic sequence
The formula for the sum of the first \(n\) terms of an arithmetic sequence is:

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

For the sum of the first 31 terms:

\[
S_{31} = \frac{31}{2} \times (2a_1 + 30d)
\]

Substitute the given value for \(S_{31}\):

\[
-3317 = \frac{31}{2} \times (2a_1 + 30d)
\]

Multiply both sides by 2 to get rid of the fraction:

\[
-6634 = 31 \times (2a_1 + 30d)
\]

Divide by 31:

\[
-214 = 2a_1 + 30d \tag{2}
\]

### Step 3: Solve the system of equations
We now have two equations:
1. \( -117 = a_1 + 16d \)
2. \( -214 = 2a_1 + 30d \)

Let’s solve these equations to find \(a_1\) and \(d\).The first term (\(a_1\)) of the arithmetic sequence is 43, and the common difference (\(d\)) is \(-10\).

Would you like further details on this solution or have any other questions?

Here are some related questions you might want to explore:
1. How is the sum formula for an arithmetic sequence derived?
2. What happens to the sequence if the common difference \(d\) changes?
3. How would you calculate the 50th term of this sequence?
4. Can an arithmetic sequence have a positive first term and a negative sum for its terms?
5. How does changing the first term affect the sum of the sequence?

**Tip:** Always use the general formulas for sequences to build equations, especially when terms and sums are given separately!

The 17th term of an arithmetic sequence is −117. The sum of the first 31 terms is −3317. Find the first term and the common difference.

Solve for the first term and common difference in an arithmetic sequence where the 17th term is −117, and the sum of the first 31 terms is −3317. Explore key formulas and theorems used to calculate these values.

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