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

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

Back

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

Front

What does a_1 represent in a geometric sequence?

Back

a_1 represents the first term of the geometric sequence.

Front

What does a_n represent in a geometric sequence?

Back

a_n represents any term in the geometric sequence.

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

Given the sequence 2, 6, 18, 54, what is the common ratio?

Back

The common ratio is 3, since each term is multiplied by 3 to get the next term.

Front

If the first term of a geometric sequence is 5 and the common ratio is 2, what are the first four terms?

Back

The first four terms are 5, 10, 20, 40.

Front

What is the next term in the sequence 1, 3, 9, 27?

Back

The next term is 81, as each term is multiplied by 3.

Front

If a geometric sequence has a common ratio of 1/2, what happens to the terms?

Back

The terms of the sequence will decrease and approach zero as n increases.

Front

What is 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 the formula: S_n = a_1 * (1 - r^n) / (1 - r) for r ≠ 1.