Understanding Sequences and Recurrence Relations

The video tutorial explains how to find the next terms in a sequence using a pattern. It introduces the sequence, identifies the pattern, and calculates the next terms. The tutorial then defines a recursive formula and uses the characteristic root technique to find a closed formula for the sequence.

What is the initial sequence given in the problem?

• 1, 3, 6, 10, 15, 21

• 2, 4, 8, 16, 32, 64

• 5, 10, 20, 40, 80, 160

• 3, 5, 11, 21, 43, 85

How is the term a sub 6 calculated?

• By adding 2 times the previous term to the current term

• By adding 2 times the term before the previous term to the previous term

• By multiplying the previous term by 2

• By subtracting the previous term from the current term

What is the recursive formula for the sequence?

• a sub n = 2 * a sub n-1 + a sub n-2

• a sub n = a sub n-1 + 2 * a sub n-2

• a sub n = a sub n-1 + a sub n-2

• a sub n = a sub n-1 - 2 * a sub n-2

What are the initial conditions required for the recursive definition?

• a sub 0 = 5, a sub 1 = 10

• a sub 0 = 3, a sub 1 = 5

• a sub 0 = 1, a sub 1 = 2

• a sub 0 = 2, a sub 1 = 4

What is the characteristic equation derived from the recurrence relation?

• x^2 + x + 2 = 0

• x^2 - x - 2 = 0

• x^2 - 2x + 1 = 0

• x^2 + 2x - 1 = 0

What are the roots of the characteristic equation?

• x = 1 and x = -2

• x = 0 and x = 3

• x = 2 and x = -1

• x = -1 and x = 1

What is the form of the solution to the recurrence relation?

• a sub n = a * n^2 + b * n

• a sub n = a * 2^n + b * (-1)^n

• a sub n = a * 3^n + b * 2^n

• a sub n = a * n + b * n^2

How is the value of 'a' determined in the closed formula?

• By solving a single equation

• By using the initial condition a sub 0

• By guessing and checking

• By solving a system of equations

What is the value of 'b' in the closed formula?

• 3/4

• 1/3

• 2/3

• 1/2

What is the final closed formula for a sub n?

• a sub n = 8/3 * 2^n + 1/3 * (-1)^n

• a sub n = 3/8 * 2^n + 1/3 * (-1)^n

• a sub n = 8/3 * n^2 + 1/3 * n

• a sub n = 3/8 * n + 1/3 * n^2