The question asks which of the given sequences have both a common difference and a common ratio, meaning the sequence is both arithmetic and geometric.

Let's analyze each sequence:

• A (4, 4, 4, 4,...):

  • Common difference: The difference between consecutive terms is 0.
  • Common ratio: The ratio between consecutive terms is 1.
  • This sequence is both arithmetic and geometric.

• B (6, -6, 6, -6, 6,...):

  • Common difference: There is no common difference since the terms alternate.
  • Common ratio: The ratio alternates between -1 and 1.
  • This is neither an arithmetic nor a geometric sequence.

• C (5, 10, 20, 40, 80,...):

  • Common difference: The difference is increasing and not constant.
  • Common ratio: The ratio between consecutive terms is 2.
  • This sequence is geometric but not arithmetic.

• D (-11, -11, -11, -11,...):

  • Common difference: The difference between consecutive terms is 0.
  • Common ratio: The ratio between consecutive terms is 1.
  • This sequence is both arithmetic and geometric.

• E (3, 7, 11, 15, 19,...):

  • Common difference: The difference between consecutive terms is 4.
  • Common ratio: There is no constant ratio.
  • This sequence is arithmetic but not geometric.

Correct Answers: A and D

Would you like more details or further explanation?

Here are 5 related questions to expand your understanding:

1. What is the difference between an arithmetic and a geometric sequence?
2. Can a sequence be neither arithmetic nor geometric?
3. How do you determine the common ratio of a geometric sequence?
4. What happens if the common ratio in a geometric sequence is negative?
5. How can you derive the general formula for the $n$-th term in an arithmetic sequence?

Tip: A sequence can be both arithmetic and geometric if either all terms are the same or each term is 0!