Exploring Geometric Sequences and Exponential Functions

This lesson covers geometric sequences as exponential functions. It explains how to identify and generate geometric sequences, relate them to exponential functions, and determine if a sequence is arithmetic, geometric, or neither. The lesson also includes methods to find the next terms in a sequence and introduces the formula for finding the nth term of a geometric sequence. Finally, it connects geometric sequences to exponential functions through real-world examples.

What is a geometric sequence?

• A sequence where each term is divided by a common ratio.

• A sequence where each term is subtracted by a constant value.

• A sequence where each term is multiplied by a common ratio.

• A sequence where each term is added by a constant value.

In a geometric sequence, what is the common ratio?

• The sum of consecutive terms.

• The product of consecutive terms.

• The difference between consecutive terms.

• The factor by which each term is multiplied to get the next term.

How do you determine if a sequence is arithmetic?

• By checking if each term is squared.

• By checking if each term is divided by a common ratio.

• By checking if each term is added or subtracted by the same amount.

• By checking if each term is multiplied by a common ratio.

What is the common difference in an arithmetic sequence?

• The product of consecutive terms.

• The difference between consecutive terms.

• The sum of consecutive terms.

• The ratio between consecutive terms.

If a sequence is neither arithmetic nor geometric, what does it mean?

• The sequence has no pattern.

• The sequence is both arithmetic and geometric.

• The sequence has a changing common ratio or difference.

• The sequence is a constant value.

How do you find the next term in a geometric sequence?

• By dividing the last term by the common ratio.

• By adding the common difference to the last term.

• By subtracting the common difference from the last term.

• By multiplying the last term by the common ratio.

What is the formula for finding the nth term of a geometric sequence?

• a_n = a_1 * r^(n-1)

• a_n = a_1 - (n-1)d

• a_n = a_1 + (n-1)d

• a_n = a_1 / r^(n-1)

In the formula a_n = a_1 * r^(n-1), what does a_1 represent?

• The nth term of the sequence.

• The common ratio.

• The first term of the sequence.

• The number of terms in the sequence.

What happens to the height of a tennis ball after each bounce if it bounces to 80% of its previous height?

• The height increases.

• The height remains the same.

• The height decreases by 80%.

• The height decreases by 20%.

If a tennis ball is dropped from a height of 12 feet and bounces to 80% of its previous height, what is the height after the first bounce?

• 6.4 feet

• 4.8 feet

• 7.2 feet

• 9.6 feet