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

Front

How do you find the common ratio in a geometric sequence?

Back

The common ratio (r) can be found by dividing any term in the sequence by the previous term: r = a_n / a_(n-1).

Front

If the first term of a geometric sequence is 4 and the common ratio is 2, what are the first four terms?

Back

The first four terms are 4, 8, 16, 32.

Front

What is the 10th term of the geometric sequence defined by a_n = 3(4)^(n-1)?

Back

The 10th term is 3 * 4^(10-1) = 3 * 262144 = 786432.

Front

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

Back

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

Front

How do you determine the first term of a geometric sequence if you know the common ratio and another term?

Back

You can rearrange the formula a_n = a_1 * r^(n-1) to find a_1: a_1 = a_n / r^(n-1).

Front

What is the 14th term of the geometric sequence defined by a_n = 2(3)^(n-1)?

Back

The 14th term is 2 * 3^(14-1) = 2 * 4782969 = 9565938.

Front

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

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.