Give the nth terms of the following: 1.) -5, 15, 35, 55, ...                 2.) 4, -1, -6, -11, ...

Let's find the \( n \)th term for both sequences:

### 1) Sequence: \(-5, 15, 35, 55, \dots\)

This is an **arithmetic sequence** where the difference between consecutive terms is constant.

**Step 1:** Identify the first term (\(a_1\)) and the common difference (\(d\)).
- \( a_1 = -5 \)
- \( d = 15 - (-5) = 20 \)

**Step 2:** Use the formula for the \(n\)th term of an arithmetic sequence:
\[
a_n = a_1 + (n-1) \times d
\]

**Step 3:** Plug in the values:
\[
a_n = -5 + (n-1) \times 20 = 20n - 25
\]

**Answer:** The \( n \)th term for the first sequence is \( \boxed{20n - 25} \).

### 2) Sequence: \( 4, -1, -6, -11, \dots \)

This is also an **arithmetic sequence**.

**Step 1:** Identify the first term (\(a_1\)) and the common difference (\(d\)).
- \( a_1 = 4 \)
- \( d = -1 - 4 = -5 \)

**Step 2:** Use the formula for the \(n\)th term of an arithmetic sequence:
\[
a_n = a_1 + (n-1) \times d
\]

**Step 3:** Plug in the values:
\[
a_n = 4 + (n-1) \times (-5) = 4 - 5n + 5 = -5n + 9
\]

**Answer:** The \( n \)th term for the second sequence is \( \boxed{-5n + 9} \).

Would you like more details or have any questions?

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**Related Questions:**

1. How do you determine if a sequence is arithmetic or geometric?
2. Can you find the sum of the first \(n\) terms in an arithmetic sequence?
3. How would the formula change if the sequence was geometric?
4. What if the common difference \(d\) was a fraction?
5. How would you find the \(n\)th term if the sequence had a common ratio instead of a difference?

**Tip:** Remember that the common difference in an arithmetic sequence can be positive, negative, or zero, depending on the sequence's progression.

Discover how to find the nth terms of arithmetic sequences: -5, 15, 35, 55 and 4, -1, -6, -11 with step-by-step solutions and formulas. This problem involves arithmetic sequence concepts and is suitable for Grades 7-9.

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