Exploring Geometric Sequences

This video tutorial covers geometric sequences, explaining how each term is found by multiplying the previous term by a common ratio. It discusses how to identify geometric sequences, calculate the common ratio, and find the next terms in a sequence. The tutorial also explains how to write explicit and recursive formulas for geometric sequences and provides examples to illustrate these concepts.

What defines a geometric sequence?

• Each term is the difference between the previous term and a constant.

• Each term is the quotient of the previous term and a constant.

• Each term is the product of the previous term and a constant.

• Each term is the sum of the previous term and a constant.

How is the common ratio of a geometric sequence calculated?

• By multiplying any term by the previous term.

• By dividing any term by the previous term.

• By subtracting any term from the previous term.

• By adding any term to the previous term.

If a sequence starts with 64 and the next term is -16, what is the common ratio?

• -4

• 4

• -1/4

• 1/4

What would be the next term in a geometric sequence after 54, if the common ratio is 1.5?

• 72

• 27

• 81

• 108

What is the common ratio if the sequence progresses from 1/3 to 2/3?

• 2

• 1

• 1/2

• 3

What is the formula to find the nth term in a geometric sequence?

• a_n = a_1 - (n-1) * r

• a_n = a_1 * r^(n-1)

• a_n = a_1 / (r^(n-1))

• a_n = a_1 + (n-1) * r

Using the nth term formula, what is the 12th term for a sequence starting with 3 and a common ratio of 2?

• 6144

• 4096

• 8192

• 2048

For a sequence with a first term of 2 and a common ratio of 3, what is the explicit formula?

• a_n = 2 * 3^(n-1)

• a_n = 2 + 3^(n-1)

• a_n = 2 / 3^(n-1)

• a_n = 2 - 3^(n-1)

What is the 12th term of a sequence where the first term is 1 and the common ratio is -3?

• 177147

• 59049

• -59049

• -177147

If a geometric sequence has a common ratio of -2, what is the sign of the 12th term if the first term is negative?

• Positive

• Negative

• Zero

• Cannot be determined