Understanding Sequences and Partial Sums

The video tutorial explores sequences a_n and b_n. It begins by analyzing sequence a_n, identifying a common difference, and deriving both recursive and closed formulas. The tutorial then introduces sequence b_n, which represents partial sums of a_n, and calculates its terms. Finally, it derives recursive and closed formulas for b_n, despite the lack of a common difference, by examining the factors of its terms.

What is the common difference in the sequence a_n?

• 2

• 3

• 4

• 5

Which formula represents the closed form of the sequence a_n?

• a_n = 3n + 1

• a_n = 4n - 3

• a_n = 5n - 2

• a_n = 2n + 5

What is b_2 in the sequence b_n?

• 5

• 6

• 8

• 7

How can b_3 be calculated using b_2?

• b_3 = b_2 + 11

• b_3 = b_2 + 9

• b_3 = b_2 + 7

• b_3 = b_2 + 5

What is the recursive formula for b_n?

• b_n = b_(n-1) + 2n

• b_n = b_(n-1) + 4n - 3

• b_n = b_(n-1) + 3n - 1

• b_n = b_(n-1) + 5n - 4

What is the initial condition for the recursive formula of b_n?

• b_2 = 8

• b_2 = 7

• b_2 = 6

• b_2 = 5

What is the first factor in the closed formula for b_n?

• n-1

• 2n

• n+1

• n

Which expression represents the second factor in the closed formula for b_n?

• n - 2

• 3n - 2

• n + 1

• 2n - 1

What is the closed formula for b_n?

• b_n = n(n + 1)

• b_n = n(3n - 2)

• b_n = n(n - 1)

• b_n = n(2n - 1)

What pattern do the second factors in the closed formula for b_n follow?

• Even numbers

• Prime numbers

• Multiples of three

• Odd numbers