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 determine if a sequence is geometric?

Back

A sequence is geometric if the ratio of consecutive terms is constant.

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

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 the sequence.

Front

Given the sequence 729, 243, 81, what is the common ratio?

Back

The common ratio is 1/3.

Front

Write the explicit formula for the geometric sequence 729, 243, 81.

Back

a_n = 729(1/3)^(n-1).

Front

Find the first four terms of the sequence defined by a_n = 4(5)^(n-1).

Back

The first four terms are 4, 20, 100, 500.

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 can be calculated using S_n = a_1 * (1 - r^n) / (1 - r) for r ≠ 1.

Front

If the first term of a geometric sequence is 100 and the common ratio is 1/2, what is the second term?

Back

The second term is 100 * (1/2) = 50.

Front

What is the relationship between the terms in a geometric sequence?

Back

Each term is obtained by multiplying the previous term by the common ratio.