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 of a geometric sequence can be calculated 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

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 does a_1 represent in a geometric sequence?

Back

a_1 represents the first term of the geometric sequence.

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 is the next term in the geometric sequence 5, 15, 45, ...?

Back

The next term is 135, found by multiplying the last term (45) by the common ratio (3).

Front

What is the common ratio in the sequence 10, 30, 90, ...?

Back

The common ratio is 3, since 30/10 = 3 and 90/30 = 3.

Front

How do you find the 5th term of the geometric sequence defined by a_n = 4 * 3^(n-1)?

Back

The 5th term is a_5 = 4 * 3^(5-1) = 4 * 81 = 324.

Front

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

Back

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

Front

If a geometric sequence has a_1 = 2 and r = 5, what is a_3?

Back

a_3 = 2 * 5^(3-1) = 2 * 25 = 50.