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 first four terms of a geometric sequence given the formula a_n = a * r^(n-1)?

Back

Substitute n = 1, 2, 3, and 4 into the formula to find the first four terms.

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 3, 15, 75, how do you find the common ratio?

Back

Divide any term by the previous term: 15/3 = 5 or 75/15 = 5. The common ratio is 5.

Front

What is the explicit formula for a geometric sequence?

Back

The explicit formula is a_n = a * r^(n-1), where a is the first term and r is the common ratio.

Front

How can you determine if a sequence is geometric?

Back

Check if the ratio of consecutive terms is constant.

Front

What is the first term of the geometric sequence defined by a_n = 4(5)^(n-1)?

Back

The first term (when n=1) is 4.

Front

If a geometric sequence has a common ratio of 2, what are the first three terms if the first term is 3?

Back

The terms are 3, 6, 12.

Front

What does it mean for a sequence to be arithmetic?

Back

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.

Front

How do you write the explicit formula for a geometric sequence with a first term of 100 and a common ratio of 1/2?

Back

The explicit formula is a_n = 100(1/2)^(n-1).