Exploring Arithmetic Sequences and Their Properties

The video introduces arithmetic sequences, which are sequences where each term is a fixed number larger than the previous one. It explains how to identify these sequences and defines them using explicit and recursive methods. Examples are provided to illustrate these concepts. The video also covers the general form of arithmetic sequences and concludes with an example of a sequence that is not arithmetic, highlighting the differences.

What is an arithmetic sequence?

• A sequence where each term increases by a different amount

• A sequence where each term is multiplied by a fixed number

• A sequence where each term is a fixed amount larger than the previous term

• A sequence where each term is a fixed amount smaller than the previous term

Which of the following is an example of an arithmetic sequence?

• 2, 4, 8, 16, 32

• 100, 90, 80, 70, 60

• 1, 3, 6, 10, 15

• -5, -3, -1, 1, 3

In the context of sequences, what does the term 'index' refer to?

• The common difference in an arithmetic sequence

• The position of a term within the sequence

• The maximum value of a term in the sequence

• The sum of all terms in the sequence

How can you define an arithmetic sequence explicitly?

• a_n = k + d(n-1)

• a_n = k * d^n

• a_n = k/d^n

• a_n = k - d(n-1)

What is the recursive definition of an arithmetic sequence?

• a_n = a_(n-1) * d

• a_n = a_(n-1) / d

• a_n = a_(n-1) + d

• a_n = a_(n-1) - d

If the first term of an arithmetic sequence is 100 and the common difference is 7, what is the third term?

• 114

• 128

• 121

• 107

What does the 'd' represent in the general formula of an arithmetic sequence?

• The first term of the sequence

• The common ratio between terms

• The common difference between terms

• The total number of terms in the sequence

How is the explicit formula for an arithmetic sequence structured?

• a_n = kn + d

• a_n = k(n-1) + d

• a_n = k + d(n-1)

• a_n = k + dn

Which sequence is not an arithmetic sequence?

• 7, 7, 7, 7

• 3, 6, 9, 12

• 1, 4, 9, 16

• 5, 10, 15, 20

What is the base case for the recursive definition of the sequence 1, 3, 6, 10?

• a_1 = 1

• a_1 = 10

• a_1 = 3

• a_1 = 6