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

What is the explicit formula for the nth term of a geometric sequence?

Back

The explicit formula 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 5, 15, 45, what is the explicit rule for the general term?

Back

a_n = 5(3)^(n-1).

Front

What is the common ratio for the sequence 28, 14, 7, 3.5?

Back

The common ratio is 1/2.

Front

Find the common ratio for the sequence 4, -16, 64, -256.

Back

The common ratio is r = -4.

Front

How do you calculate the 12th term of the sequence defined by a_n = 6(2)^(n-1)?

Back

To find a_12, substitute n=12: a_12 = 6(2)^(12-1) = 12,288.

Front

What does the term 'explicit rule' mean in the context of sequences?

Back

An explicit rule provides a formula to calculate the nth term of a sequence directly without needing to know the previous terms.

Front

What is the first term of the sequence represented by the equation 2(7)^(n-1)?

Back

The first term (when n=1) is 2(7)^(1-1) = 2.

Front

If the common ratio is negative, what effect does it have on the sequence?

Back

A negative common ratio causes the terms of the sequence to alternate in sign.