Question

The first 8 terms of the geometric sequence $$ -12,-48,-192,-768, \ldots . $$

Short answer

The first 8 terms of the series are -12, -48, -192, -768, -3072, -12288, -49152 and -196608.

Step-by-step solution

Find the common ratio

Divide the second term by the first term to find the common ratio \(r\). In this case, \(-48 / -12 = 4\), so \(r = 4\).

Use the formula to find the terms

The formula is \(a_n = a_1*r^{(n-1)}\). Here, \(a_1 = -12\), \(r = 4\), \(n\) is the desired term. By substituting the known values in the formula for each desired term, the first 8 terms of the sequence are -12, -48, -192, -768, -3072, -12288, -49152, and -196608.

Key concepts

Common Ratio

Understanding the concept of the common ratio is crucial when dealing with geometric sequences. A geometric sequence is a series 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. This ratio remains constant throughout the sequence.

For example, consider the geometric sequence presented in the exercise: \[-12, -48, -192, -768, \text{...}\] To identify the common ratio (\(r\)), we divide any term by the term preceding it. In this case, \(-48 \/ -12 = 4\), confirming that the common ratio is 4. This consistency of the common ratio allows us to predict subsequent terms and understand the pattern within the sequence.

Sequence Formula

The sequence formula is foundational for finding any term within a geometric sequence. The general formula for the \(n\)-th term of a geometric sequence is: \[a_n = a_1 * r^{(n-1)}\]
Where:

• \(a_n\) is the \(n\)-th term of the sequence,
• \(a_1\) is the first term,
• \(r\) is the common ratio, and
• \(n\) is the term number we want to find.

In the context of our exercise, using this formula allows us to calculate each term efficiently. Given \(a_1 = -12\) and \(r = 4\), you can find any term of the sequence by just substituting the values and exponentiating the common ratio to the appropriate power, demonstrating the utility and power of this formula in handling geometric sequences.

Terms of a Sequence

The individual elements of a sequence are called terms. Each term can be found by using the sequence formula mentioned previously. In a geometric sequence, it's particularly straightforward to find any term if you know the first term and the common ratio.

In our exercise, the first eight terms are derived sequentially by applying the formula \(a_n = -12*4^{(n-1)}\). Starting with \(n = 1\) for the first term and incrementing \(n\) for each subsequent term, we generate the progression:

• \(1\)-st term: \(a_1 = -12\)
• \(2\)-nd term: \(a_2 = -12 * 4\)
• \(3\)-rd term: \(a_3 = -12 * 4^2\)
• ... and so forth, up to the \(8\)-th term.

This illustrates how with a grasp of the initial term and the common ratio, one can easily compute any term in the sequence.