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

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

Is the sequence -432, 72, -12, 2,... geometric?

Back

Yes, this sequence is geometric because each term is obtained by multiplying the previous term by a common ratio of -1/6.

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

Given the sequence 125, 25, 5,..., write the rule a_n using the geometric sequence formula.

Back

a_n = (125)(1/5)^(n-1).

Front

What is the common ratio of the sequence 3, 6, 12, 24,...?

Back

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

Front

If a geometric sequence has a first term of 4 and a common ratio of 3, what is the 5th term?

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 is given by S_n = a_1 * (1 - r^n) / (1 - r) for r ≠ 1.

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 common ratio for the geometric sequence 2, 6, 18, 54,...?

Back

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