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 by the previous term: r = a_n / a_(n-1).

Front

What is the explicit formula for a geometric sequence?

Back

The explicit formula for a geometric sequence is a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.

Front

Given the sequence 3, -12, 48, -192, what is the common ratio?

Back

The common ratio is -4.

Front

What is the 9th term of the geometric sequence 4, 8, 16, 32?

Back

The 9th term is 1,024.

Front

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

Back

To calculate the nth term, use the formula a_n = a_1 * r^(n-1).

Front

What is the common ratio for the sequence 28, 14, 7, 3.5?

Back

The common ratio is 1/2.

Front

If a geometric sequence has a first term of 6 and a common ratio of 2, what is the 12th term?

Back

The 12th term is 12,288.

Front

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

Back

The sum S_n of the first n terms is given by S_n = a_1 * (1 - r^n) / (1 - r) for r ≠ 1.

Front

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

Back

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