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 calculated using the formula: a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.

Front

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

Back

The common ratio (r) can be found by dividing any term by the previous term: r = a_n / a_(n-1).

Front

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

Back

The sum (S_n) of the first n terms of a geometric sequence can be calculated using the formula: S_n = a_1 * (1 - r^n) / (1 - r), for r ≠ 1.

Front

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

Back

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

Front

If the first term of a geometric sequence is 5 and the common ratio is 2, what is the 4th term?

Back

The 4th term is 40, calculated as 5 * 2^(4-1) = 5 * 8 = 40.

Front

What are the next three terms in the geometric sequence 4, 12, 36, ...?

Back

The next three terms are 108, 324, and 972, with a common ratio of 3.

Front

Evaluate the sum of the first 5 terms of the geometric sequence 1, 3, 9, ...

Back

The sum is 121, calculated as S_5 = 1 * (1 - 3^5) / (1 - 3) = 121.

Front

What is the 6th term of the geometric sequence with a_1 = 2 and r = 5?

Back

The 6th term is 31250, calculated as 2 * 5^(6-1) = 2 * 3125 = 6250.

Front

If a geometric sequence has a common ratio of 1/2, what happens to the terms?

Back

The terms will decrease and approach zero as they are multiplied by 1/2.