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 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

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 and r is the common ratio.

Front

What is the explicit formula for a geometric sequence?

Back

The explicit formula for a geometric sequence is 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 -4, what is the 3rd term?

Back

The 3rd term is a_3 = 3 * (-4)^(3-1) = 3 * 16 = 48.

Front

What is the 5th term of the geometric sequence 2, 6, 18, ...?

Back

The 5th term is a_5 = 2 * 3^(5-1) = 2 * 81 = 162.

Front

How do you determine if a sequence is geometric?

Back

A sequence is geometric if the ratio of consecutive terms is constant.

Front

What is the formula to find the sum of the first n terms of a geometric sequence?

Back

The sum of the first n terms of a geometric sequence is given by S_n = a_1 * (1 - r^n) / (1 - r), where r is not equal to 1.

Front

If a geometric sequence has a common ratio of 1/2 and the first term is 8, what is the 4th term?

Back

The 4th term is a_4 = 8 * (1/2)^(4-1) = 8 * (1/8) = 1.

Front

What is the relationship between the common ratio and the terms of a geometric sequence?

Back

The common ratio determines how each term in the sequence is related to the previous term; it can be positive or negative.