Question

Are the sequences geometric? For those that are, give a formula for the \(n^{\text {th }}\) term. $$ -2,4,-8,16, \ldots $$

Short answer

If so, what is the formula for the nth term of the sequence? Answer: Yes, the sequence is geometric. The formula for the nth term of the sequence is \(a_n = -2 \cdot (-2)^{(n-1)}\).

Step-by-step solution

Check if the sequence is geometric

To check if the given sequence is geometric, we need to verify that the ratio between consecutive terms (the common ratio) is the same for all pairs of terms. Let's calculate the ratio for the first two terms and then compare it with the ratio for the next pair of terms: Ratio between the first and second term: \(\frac{4}{-2} = -2\) Ratio between the second and third term: \(\frac{-8}{4} = -2\) Since the ratio is the same for both pairs of terms, we can conclude that the sequence is, in fact, geometric with a common ratio of -2.

Finding the formula for the nth term

In a geometric sequence, the general formula to find the nth term is given by: \(a_n = a_1 \cdot r^{(n-1)}\) where \(a_n\) is the nth term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number. In our given sequence, we have the first term \(a_1 = -2\) and the common ratio \(r = -2\). Now, we can plug in these values into our formula to find the general formula for the nth term: \(a_n = -2 \cdot (-2)^{(n-1)}\) This is the formula for the nth term of this geometric sequence.

Key concepts

Common Ratio

A geometric sequence is a type of sequence where the ratio between any two consecutive terms remains constant. This constant is called the "common ratio." To determine whether a sequence is geometric, you simply need to check if this ratio is consistent throughout the sequence.
In the sequence given in the exercise

• -2, 4, -8, 16,...

to find the common ratio, divide the second term by the first term:
\[\frac{4}{-2} = -2\].
Then, check the next pairs:
\[\frac{-8}{4} = -2\]. Since these ratios are equal, each pair of terms has the same common ratio of \(-2\), confirming it's a geometric sequence.
The common ratio plays a crucial role because it tells us the factor by which each term increases or decreases as the sequence progresses. But it's also an indicator of the sequence's growth pattern, helping us understand how quickly or slowly it evolves.

Nth Term Formula

The nth term formula in a geometric sequence provides a way to find any term in that sequence without listing all preceding terms.
The general formula is given by:
\[a_n = a_1 \cdot r^{(n-1)}\],
where:

• \(a_n\) is the nth term,
• \(a_1\) is the first term,
• \(r\) is the common ratio,
• \(n\) is the term number.

For our sequence (\(-2, 4, -8, 16,\ldots\)), we substitute \(a_1 = -2\) and \(r = -2\) into the formula:
\[a_n = -2 \cdot (-2)^{(n-1)}\].
This formula is powerful because it allows you to calculate any term directly, without computation or comparison of other terms.
Whether you're finding the 5th term or the 50th, the nth term formula gives you results instantly by substituting the desired term number \(n\).

Sequence Analysis

Analyzing a geometric sequence involves understanding its characteristics and behavior. With a common ratio of \(-2\) in our exercise, it's clear that the sequence alternates in sign and grows quickly in both positive and negative directions.
Each successive term is the product of the previous term and \(-2\), which flips its sign and doubles its absolute value.
To conduct a thorough analysis:

• Observe the pattern of signs and magnitudes: the sequence goes from negative to positive and back, increasing exponentially.
• Consider potential applications: Understanding the sequence's behavior helps in fields like finance for compound interest or physics for wave amplitudes.
• Predict future terms or determine characteristics like bounds: By using the nth term formula, anticipate the sequence's shape over time.

Recognizing these patterns can deepen comprehension and provide insights into real-world phenomena, where such mathematical sequences frequently occur.