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 in the sequence by the previous term (r = a_n / a_(n-1)).

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 and r is the common ratio.

Front

If the first term of a geometric sequence is 3 and the common ratio is 2, what is the 5th term?

Back

The 5th term is a_5 = 3 * 2^(5-1) = 3 * 16 = 48.

Front

What does the term 'common ratio' refer to in a geometric sequence?

Back

The common ratio is the constant factor between consecutive terms in a geometric sequence.

Front

Can a geometric sequence have a common ratio of zero?

Back

No, a geometric sequence cannot have a common ratio of zero, as this would not allow for a valid sequence of terms.

Front

What is the difference between an arithmetic sequence and a geometric sequence?

Back

An arithmetic sequence adds a constant value (common difference) to each term, while a geometric sequence multiplies each term by a constant value (common ratio).

Front

If a geometric sequence has a common ratio of -1/4, what is the second term if the first term is 64?

Back

The second term is 64 * (-1/4) = -16.

Front

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

Back

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

Front

What happens to the terms of a geometric sequence if the common ratio is greater than 1?

Back

If the common ratio is greater than 1, the terms of the geometric sequence will increase without bound.