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

If the first term of a geometric sequence is 5 and the common ratio is 3, what is the 4th term?

Back

The 4th term is a_4 = 5 * 3^(4-1) = 5 * 27 = 135.

Front

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

Back

The common ratio is 3.

Front

What is the next term in the geometric sequence: 1, 2, 4, 8...?

Back

The next term is 16.

Front

If a geometric sequence has a common ratio of 1/2 and the first term is 16, what is the 5th term?

Back

The 5th term is a_5 = 16 * (1/2)^(5-1) = 16 * (1/16) = 1.

Front

What is 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 negative?

Back

If the common ratio is negative, the terms of the sequence will alternate in sign.

Front

What is the common ratio for the sequence: 81, 27, 9, 3...?

Back

The common ratio is 1/3.