Introduction to Geometric Sequences: The Common Ratio

Foundational

Discover the core concept of a geometric sequence. This stack focuses on identifying these sequences by finding their constant 'common ratio' (r) and understanding the pattern of multiplication.

Introduction to Geometric Sequences: The Common Ratio

1 of 12

What is the defining characteristic of a geometric sequence?

A geometric sequence is a sequence in which the ratio between any two consecutive terms is constant.

What is the 'common ratio' of a geometric sequence, and what is its symbol?

The common ratio is the constant value that each term is multiplied by to get the next term. It is denoted by the variable \(r\).

How do you calculate the common ratio, \(r\), for a geometric sequence?

Divide any term by its preceding term: \(r = \frac{a_{n+1}}{a_n}\).

Is the sequence \(3, 6, 12, 24, ...\) geometric? If so, what is its common ratio?

Yes, it is a geometric sequence. The common ratio is \(r = \frac{6}{3} = 2\).

Is the sequence \(80, 40, 20, 10, ...\) geometric? If so, what is its common ratio?

Yes, it is a geometric sequence. The common ratio is \(r = \frac{40}{80} = \frac{1}{2}\) or \(0.5\).

Is the sequence \(5, -10, 20, -40, ...\) geometric? If so, what is its common ratio?

Yes, it is a geometric sequence. The common ratio is \(r = \frac{-10}{5} = -2\).

Is the sequence of perfect squares, \(1, 4, 9, 16, ...\), a geometric sequence?

No. The ratio between consecutive terms is not constant (\(\frac{4}{1}=4\), but \(\frac{9}{4}=2.25\)).

If a geometric sequence has \(a_1 = 4\) and a common ratio of \(r = 3\), what are the first four terms?

The first four terms are \(4\), \(12\), \(36\), and \(108\).

If a geometric sequence has \(a_1 = 64\) and \(r = -\frac{1}{2}\), what are the first four terms?

The first four terms are \(64\), \(-32\), \(16\), and \(-8\).

The fourth term of a geometric sequence is \(18\) and the fifth term is \(54\). What is the common ratio, \(r\)?

The common ratio is \(r = \frac{a_5}{a_4} = \frac{54}{18} = 3\).

What happens to the terms of a geometric sequence when the absolute value of the common ratio is less than 1 (\(|r| < 1\))?

The terms get progressively closer to zero as \(n\) increases. The sequence converges to zero.

What is the effect of a negative common ratio (\(r < 0\)) on the terms of a geometric sequence?

The signs of the terms will alternate (e.g., positive, negative, positive, negative, ...).

Introduction to Geometric Sequences: The Common Ratio

Confirm Action

Prompt