Exploring Explicit and Recursive Formulas in Geometric Sequences

The video tutorial explains a sequence defined by a function G, where each term is half of the previous term, forming a geometric series. It covers how to define the sequence explicitly and recursively, using algebraic manipulation to show equivalent forms. The tutorial encourages viewers to think about the sequence's properties and how to express it mathematically.

What is the first term of the sequence given in the video?

• 42

• 168

• 21

• 84

How do you go from the first term to the second term in the sequence?

• Add 84

• Multiply by 1/2

• Subtract 84

• Divide by 2

What type of sequence is described in the video?

• Geometric

• Arithmetic

• Fibonacci

• Harmonic

What is the common ratio in the geometric series described?

• 1/2

• 1/4

• 2

• 1/3

How can the explicit formula for G(n) be written?

• 168 * (1/2)^(n-1)

• 168 - (1/2)^(n-1)

• 168 + (1/2)^(n-1)

• 168 / (1/2)^(n-1)

What is the value of G(3) in the sequence?

• 42

• 84

• 21

• 168

How can the explicit formula be algebraically manipulated?

• 168 / 2^(n-1)

• 168 * 2^(n-1)

• 168 + 2^(n-1)

• 168 - 2^(n-1)

What is the recursive definition for G(n) when n > 1?

• G(n) = G(n-1) + 1/2

• G(n) = 2 * G(n-1)

• G(n) = G(n-1) - 1/2

• G(n) = 1/2 * G(n-1)

What is the initial value for G(1) in the recursive definition?

• 168

• 84

• 21

• 42

How can you verify the recursive definition for G(2)?

• G(2) = 2 * G(1)

• G(2) = 1/2 * G(1)

• G(2) = G(1) + 1/2

• G(2) = G(1) - 1/2