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.