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 = a₁ + (n - 1)d, where a₁ 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 x (2a₁ + (n - 1)d) or S_n = n/2 x (a₁ + a_n), where a_n is the nth term.

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

The first term a₁ = 5 and the common difference d = 4. 
Using the formula 
a_n = a₁ + (n - 1)d, we find a₂₀ = 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 a₁ is the first term and r is the common ratio.

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

The first term a₁ = 4 and the common ratio r = 3. Using the formula 
a_n = a_1.r^{(n - 1)}, 
we find 
a₅ = 4. 3^{(5 - 1)}
= 4 x 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 = a₁ * (1 - rⁿ) / (1 - r) for r ≠ 1.

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

The first term 
a₁ = 1 and the common ratio r = 2. 
The sum is 
S₄ = 1 * (1 - 2⁴) / (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.