Geometric Series

Front

What is a geometric series?

Back

A geometric series is the sum of the terms of a geometric sequence, 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

What is the common ratio in a geometric sequence?

Back

The common ratio (r) is the factor by which we multiply each term to get the next term in a geometric sequence.

Front

Identify the common ratio in the sequence: 3, 6, 12, 24, ...

Back

The common ratio is 2, since each term is multiplied by 2 to get the next term.

Front

What is the first term of the geometric sequence defined by a_n = 5(3)^(n-1)?

Back

The first term (a_1) is 5, when n = 1.

Front

Find the first four terms of the sequence given by a_n = 2(4)^(n-1).

Back

The first four terms are 2, 8, 32, 128.

Front

What is the sum of the first n terms of a geometric series?

Back

The sum S_n of the first n terms of a geometric series can be calculated using the formula: S_n = a_1 * (1 - r^n) / (1 - r), where a_1 is the first term and r is the common ratio.

Front

What is the sum of the first 5 terms of the geometric series with a_1 = 2 and r = 3?

Back

The sum is S_5 = 2 * (1 - 3^5) / (1 - 3) = 2 * (1 - 243) / (-2) = 242.

Front

What type of sequence is 400, 200, 100, 50, 25, ...?

Back

It is a geometric sequence with a common ratio of 1/2.

Front

If a geometric sequence has a first term of 10 and a common ratio of 2, what is the 4th term?

Back

The 4th term is a_4 = 10 * 2^(4-1) = 10 * 8 = 80.