Exploring Sequences and Series

What is the nth term formula for an arithmetic sequence?

• a_n = a + n * d

• a_n = a + (d/n)

• a_n = a + (n - 1) * d

• a_n = a * n + d

If the first term of an arithmetic sequence is 5 and the common difference is 3, what is the 10th term?

• 25

• 20

• 32

• 15

Determine the sum of the first 20 terms of the arithmetic sequence: 2, 5, 8, ...

• 720

• 500

• 610

• 450

What is the common ratio in the geometric sequence: 3, 9, 27, ...?

• 3

• 12

• 9

• 6

Find the 6th term of the geometric sequence where the first term is 4 and the common ratio is 2.

• 128

• 256

• 64

• 32

What is the formula for the sum of the first n terms of a geometric series?

• S_n = a(r^n - 1) / (r - 1)

• S_n = a(nr) / (1 - r)

• S_n = a(1 + r^n) / (1 + r)

• S_n = a(1 - r^n) / (1 - r)

If a geometric sequence has a first term of 1 and a common ratio of 1/2, what is the sum of the first 5 terms?

• 15/8

• 1/2

• 7/4

• 31/16

Define a recursive sequence and provide an example.

• Harmonic sequence: H(n) = 1/n with H(1) = 1.

• Geometric sequence: G(n) = G(n-1) * r with G(0) = 1 and r = 3.

• Fibonacci sequence: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

• Arithmetic sequence: a(n) = a(n-1) + d with a(0) = 1 and d = 2.

What is the recursive formula for the Fibonacci sequence?

• F(n) = F(n-1) + F(n-2) for n > 1, with F(0) = 0 and F(1) = 1.

• F(n) = F(n-1) + 2F(n-2) for n > 1

• F(n) = F(n-1) * F(n-2) for n > 1

• F(n) = F(n-1) - F(n-2) for n > 1

Explain the difference between convergence and divergence in series.

• Convergence refers to a series approaching a finite limit, while divergence means the series does not approach a finite limit.

• Convergence refers to a series that diverges, while divergence means it converges to a limit.

• Convergence indicates a series oscillates between values, while divergence means it stabilizes at a point.

• Convergence means a series grows indefinitely, while divergence means it approaches zero.