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 determine if a sequence is geometric?

Back

A sequence is geometric if the ratio of any term to its preceding term is constant.

Front

What is the common ratio in a geometric sequence?

Back

The common ratio is the factor by which we multiply each term to get the next term in the sequence.

Front

If the first term of a geometric sequence is 3 and the common ratio is 2, what are the first four terms?

Back

3, 6, 12, 24.

Front

What is the explicit formula for a geometric sequence?

Back

The explicit formula is 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 5th term of the geometric sequence 2, 6, 18, 54...?

Back

The 5th term is 162.

Front

How do you find the nth term of a geometric sequence?

Back

Use the formula a_n = a_1 * r^(n-1), substituting the values for a_1 and r.

Front

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

Back

The sum S_n = a_1 * (1 - r^n) / (1 - r) for r ≠ 1.

Front

If a geometric sequence has a common ratio of 1/2 and the first term is 8, what is the 4th term?

Back

The 4th term is 1.

Front

What is the relationship between geometric sequences and exponential functions?

Back

Geometric sequences can be represented by exponential functions, as each term can be expressed as a constant multiplied by a power of the common ratio.