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 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 5 and the common ratio is 3, what is the second term?

Back

The second term is 15 (5 * 3).

Front

What does a₁ represent in a geometric sequence?

Back

a₁ represents the first term of the sequence.

Front

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

Back

The nth term can be found using the formula a_n = a₁ * r^n-1, where a₁ is the first term and r is the common ratio.

Front

Is the sequence 3, 6, 12, 24 a geometric sequence?

Back

Yes, it is a geometric sequence with a common ratio of 2.

Front

What is the explicit formula for the geometric sequence 4, 12, 36, ...?

Back

The explicit formula is a_n = 4 * 3^n-1.

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 sum of the first n terms of a geometric sequence?

Back

The sum can be calculated using the formula S_n = a₁ * (1 - rⁿ) / (1 - r) for r ≠ 1.

Front

Can a geometric sequence have a negative common ratio?

Back

Yes, a geometric sequence can have a negative common ratio, which will cause the terms to alternate in sign.