Problem 11 - Algebra 2

Question

Is the sequence geometric? If so, find the common ratio and the next two terms. $$ -1,-6,-36,-216, \dots $$

Step-by-Step Solution

Answer

Yes, the sequence is geometric with a common ratio of 6. The next two terms are -1296 and -7776.

Step 1: Check if sequence is Geometric

Divide successive terms to see if all the ratios equate. For instance, divide \(-6\) by \(-1\), then \(-36\) by \(-6\) and so on to see if the ratios are equal

Step 2: Find the Common Ratio

If step 1 indicates that the sequence is geometric, the common ratio is the value from the division. In this case, divide \(-6\) by \(-1\) to obtain the common ratio

Step 3: Find the next terms

Multiply the last term in the sequence by the common ratio. For instance, for the next term, multiply \(-216\) by the common ratio. Repeat again to get the term after.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a crucial element that determines the multiplicative relationship between consecutive terms. To find this ratio, simply divide one term by its preceding term. For example, in the given sequence

• The first term is \(-1\).
• The second term is \(-6\).

To find the common ratio, \(-6\) is divided by \(-1\), resulting in a value of \(r = 6\). This pattern is consistent throughout the sequence:

• \(-36/-6 = 6\)
• \(-216/-36 = 6\)

This confirms that the sequence is geometric with a consistent common ratio of \(6\). Understanding the common ratio is essential, enabling us to predict future terms in the sequence reliably.

Sequence Analysis

Analyzing a sequence involves examining its structure and pattern. For geometric sequences, the primary focus is on identifying whether the ratios between successive terms are constant.

For the sequence \(-1, -6, -36, -216, ...\), we can analyze it as:

• The ratio between \(-6\) and \(-1\) is \(6\).
• The ratio between \(-36\) and \(-6\) is also \(6\).
• The ratio between \(-216\) and \(-36\) remains \(6\).

Since the ratios are consistent, we confirm it's a geometric sequence, differing from other types like arithmetic sequences, which rely on a constant difference rather than a ratio. Recognizing this pattern allows you to classify the type of sequence correctly and apply relevant methods to predict and analyze it.

Next Terms Determination

To determine the next terms in a geometric sequence, use the common ratio to multiply the last known term. This straightforward approach allows you to extend the sequence with confidence.

Given the sequence \(-1, -6, -36, -216, ...\), and knowing the common ratio \(r = 6\), we calculate:

• Multiply \(-216\) (the last term) by \(6\) to get \(-1296\).
• Then multiply \(-1296\) by \(6\) to get \(-7776\).

This method reflects how geometric sequences inherently grow, or shrink, exponentially, driven by their consistent multiplicative factor. Mastering this approach ensures you can effortlessly compute further terms in any geometric sequence.