Exploring Geometric Sequences and Their Explicit Formulas

The video tutorial explains the concept of geometric sequences and how to use explicit formulas to find specific terms in a sequence. It covers several examples, including sequences with positive, negative, and fractional common ratios, as well as real-world applications like car depreciation. The tutorial also demonstrates converting recursive formulas to explicit ones, emphasizing the importance of the first term and common ratio in constructing these formulas.

What is a geometric sequence?

• A sequence where the sum between terms is constant

• A sequence where each term is the square root of the previous term

• A sequence where the ratio between terms is constant

• A sequence where each term is the square of the previous term

What does the explicit formula for a geometric sequence allow us to do?

• Predict the possible future terms without any calculations

• Calculate the sum of all terms up to the nth term

• Find any term in the sequence without computing all previous terms

• Determine the common ratio by knowing the first term

How do you find a specific term in a geometric sequence using the explicit formula?

• Multiply the first term by the common ratio raised to the power of (n-1)

• Add the common ratio to the first term n times

• Divide the first term by the common ratio raised to the power of n

• Subtract the common ratio from the first term n times

If the first term of a geometric sequence is 4 and the common ratio is 2, what is the eighth term?

• 2048

• 1024

• 512

• 256

For a sequence with a first term of 486 and a common ratio of 1/3, what is the sixth term?

• 6

• 54

• 2

• 18

What is the common ratio if the sequence terms are 5, -25, 125, -625?

• -25

• 5

• -5

• 25

What is the value of a car after 8 years if it depreciates at a rate of 20% per year from an initial value of $24,000?

• $5,033.16

• $3,200.00

• $4,800.00

• $6,000.00

If a sequence starts at -64 and each term halves the previous, what is the fifth term?

• -8

• -32

• -4

• -16

What is the explicit formula for a sequence starting at 1000 with a common ratio of 1/5?

• 1000 * (1/5)^(n-1)

• 1000 - (1/5)^n

• 1000 / (1/5)^(n-1)

• 1000 + (1/5)^n

What is the seventh term of a sequence with a first term of 1000 and a common ratio of 1/5?

• 64

• 32

• 128

• 256