Geometric sequences

Front

What is a geometric sequence?

Back

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Front

What is the common ratio in a geometric sequence?

Back

The common ratio is the factor by which we multiply each term to get the next term in a geometric sequence.

Front

How do you find the nth term of a geometric sequence?

Back

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, r is the common ratio, and n is the term number.

Front

What is the first term of a geometric sequence?

Back

The first term of a geometric sequence is the initial value from which the sequence starts, denoted as a_1.

Front

If the first term of a geometric sequence is 3 and the common ratio is 2, what is the second term?

Back

The second term is 3 * 2 = 6.

Front

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

Back

The sum of the first n terms (S_n) of a geometric sequence can be calculated using the formula: S_n = a_1 * (1 - r^n) / (1 - r), where r is not equal to 1.

Front

What happens to a geometric sequence if the common ratio is less than 1?

Back

If the common ratio is less than 1, the terms of the geometric sequence will decrease and approach zero.

Front

What is an example of a geometric sequence?

Back

An example of a geometric sequence is 2, 6, 18, 54, ... where each term is multiplied by 3.

Front

What is the difference between an arithmetic sequence and a geometric sequence?

Back

In an arithmetic sequence, each term is added by a constant (common difference), while in a geometric sequence, each term is multiplied by a constant (common ratio).

Front

If a geometric sequence has a common ratio of -2, what is the sign of the terms?

Back

The terms will alternate in sign, starting with the sign of the first term.