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 explicit formula for a geometric sequence?

Back

The explicit formula for a geometric sequence is given by a_n = a_1 * r^(n-1), where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the term number.

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

How do you find the nth term of a geometric sequence?

Back

To find the nth term of a geometric sequence, use the formula a_n = a_1 * r^(n-1), substituting the values of a_1, r, and n.

Front

What is the first term of the geometric sequence 2, 6, 18, 54, ...?

Back

The first term (a_1) is 2.

Front

What is the common ratio of the geometric sequence 2, 6, 18, 54, ...?

Back

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

Front

Determine if the sequence 5, 15, 45, 135, ... is arithmetic, geometric, or neither.

Back

The sequence is geometric.

Front

Determine if the sequence 10, 20, 30, 40, ... is arithmetic, geometric, or neither.

Back

The sequence is arithmetic.

Front

Find a_5 in the geometric sequence 3, 9, 27, ...

Back

a_5 = 3 * 3^(5-1) = 243.

Front

What is the explicit formula for the geometric sequence with a_1 = 4 and r = 2?

Back

a_n = 4 * 2^(n-1).