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 formula for the nth term of 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

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

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 a_4 = 3 * 2^(4-1) = 3 * 8 = 24.

Front

What is the explicit formula for the geometric sequence 729, 243, 81?

Back

The explicit formula is a_n = 729(1/3)^(n-1).

Front

Is the sequence 4, 12, 36, 108 geometric?

Back

Yes, it is geometric because each term is multiplied by the common ratio of 3.

Front

What is the common ratio of the sequence 5, 15, 45, 135?

Back

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

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 n increases.

Front

What is the first term of a geometric sequence if the 5th term is 48 and the common ratio is 2?

Back

The first term is a_1 = 48 / (2^(5-1)) = 48 / 16 = 3.

Front

How can you determine if a sequence is geometric?

Back

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