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

How do you find the common ratio in a geometric sequence?

Back

The common ratio (r) can be found by dividing any term in the sequence by the previous term (r = a_n / a_(n-1)).

Front

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

Back

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

Front

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

Back

The first four terms are 3, 6, 12, 24.

Front

What is the common ratio of the sequence 2, 6, 18, 54, ...?

Back

The common ratio is 3.

Front

How do you determine the next terms in a geometric sequence?

Back

To find the next terms, multiply the last known term by the common ratio.

Front

What is the common ratio for the sequence 5, -10, 20, -40, ...?

Back

The common ratio is -2.

Front

Given the sequence 4, 12, 36, ..., what is the common ratio?

Back

The common ratio is 3.

Front

If a geometric sequence has a common ratio of 1/2 and the first term is 8, what are the next three terms?

Back

The next three terms are 4, 2, 1.

Front

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

Back

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