What defines an arithmetic sequence?

An arithmetic sequence is defined by a constant difference between consecutive terms. The nth term can be expressed as a_n = a_1 + (n - 1)d, where a_1 is the first term and d is the common difference.

What is the formula for the sum of the first n terms of an arithmetic sequence?

The sum of the first n terms (S_n) can be calculated using the formula: S_n = n/2 * (2a_1 + (n - 1)d) or S_n = n/2 * (a_1 + a_n), where a_n is the nth term.

In the arithmetic sequence 5, 9, 13, ..., what is the 20th term?

The first term a_1 = 5 and the common difference d = 4. Using the formula a_n = a_1 + (n - 1)d, we find a_20 = 5 + (20 - 1) * 4 = 5 + 76 = 81.

What is a geometric sequence?

A geometric sequence is defined by each term being obtained by multiplying the previous term by a fixed non-zero number called the common ratio (r). For example, in the sequence 3, 6, 12, ..., the common ratio is 2.

What is the formula for the nth term of a geometric sequence?

The nth term of a geometric sequence can be found using the formula: a_n = a_1 * r^(n - 1), where a_1 is the first term and r is the common ratio.

Calculate the 5th term of the geometric sequence 4, 12, 36, ...

The first term a_1 = 4 and the common ratio r = 3. Using the formula a_n = a_1 * r^(n - 1), we find a_5 = 4 * 3^(5 - 1) = 4 * 81 = 324.

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

The sum of the first n terms (S_n) of a geometric sequence can be calculated using S_n = a_1 * (1 - r^n) / (1 - r) for r ≠ 1.

Find the sum of the first 4 terms of the geometric sequence 1, 2, 4, ...

The first term a_1 = 1 and the common ratio r = 2. The sum is S_4 = 1 * (1 - 2^4) / (1 - 2) = 1 * (1 - 16) / (-1) = 15.

What is the difference between a finite and an infinite sequence?

A finite sequence has a specific number of terms, while an infinite sequence continues indefinitely without an endpoint. For example, the sequence of natural numbers is infinite.

Identify the sequence type: 1, 1/2, 1/4, 1/8, ...

This is a geometric sequence where the common ratio r = 1/2. Each term is obtained by multiplying the previous term by 1/2.