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, i.e., r = a_n / a_(n-1).

Front

Given the geometric sequence -250, -50, -10, 2, what is the common ratio?

Back

r = 1/5.

Front

Is the sequence 2, 4, 12, 48 a geometric sequence?

Back

No, because the ratio between consecutive terms is not constant.

Front

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

Back

The nth term can be found using the formula: a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.

Front

Find the 9th term in the geometric sequence 4, 8, 16, 32.

Back

The 9th term is 1,024.

Front

What is the explicit formula for the geometric sequence 3, 24, 192, 1536?

Back

The explicit formula is a_n = 3 * 8^(n-1).

Front

Write the explicit rule for the geometric sequence 2, -12, 72.

Back

The explicit rule is a_n = 2 * (-6)^(n-1).

Front

What is the formula for 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 is the common ratio of the geometric sequence 5, 15, 45, 135?

Back

The common ratio is r = 3.