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 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

If the first term of a geometric sequence is 3 and the common ratio is 2, what is the 4th term?

Back

The 4th term is 3 * 2^(4-1) = 3 * 8 = 24.

Front

What does 'r' represent in a geometric sequence?

Back

In a geometric sequence, 'r' represents the common ratio, which is the factor by which we multiply each term to get the next term.

Front

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

Back

To find the common ratio, divide any term by the previous term (r = a_n / a_(n-1)).

Front

What is the common ratio for the sequence: 5, 15, 45, ...?

Back

The common ratio is 3, since 15/5 = 3 and 45/15 = 3.

Front

What does a_1 represent in a geometric sequence?

Back

a_1 represents the first term of the geometric sequence.

Front

If a geometric sequence has a common ratio of 1/3 and the first term is 9, what is the 5th term?

Back

The 5th term is 9 * (1/3)^(5-1) = 9 * (1/3)^4 = 9 * 1/81 = 1/9.

Front

What is the next term in the geometric sequence 2, 6, 18, ...?

Back

The next term is 54, since the common ratio is 3 (6/2 = 3 and 18/6 = 3).

Front

In a geometric sequence, if the common ratio is negative, what happens to the terms?

Back

If the common ratio is negative, the terms will alternate in sign (positive, negative, positive, etc.).