Problem 10 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio.Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio. $$4,19,34,49, \dots$$

Step-by-Step Solution

Answer

The given sequence (4,19,34,49) is not a geometric sequence.

Step 1: Compute The Ratios Between Consecutive Terms

First of all, calculate the ratios between consecutive terms. Beginning from the second term, divide each term by its preceding term. For given sequence, \(4,19,34,49\), these ratios are \(19/4 = 4.75\), \(34/19 \approx 1.79\), and \(49/34 \approx 1.44\).

Step 2: Check the Consistency Of The Ratios

Next step, check if these ratios are identical. In our case, the computed ratios are 4.75, 1.79 and 1.44, which are not equal.

Step 3: Drawing Conclusion

Since the ratios between consecutive terms are not equal, we can conclude that the sequence is not a geometric sequence.

Key Concepts

Common Ratio

When studying sequences, identifying if a sequence is geometric revolves around finding what's called the "common ratio." But what exactly is a common ratio? In a geometric sequence, the common ratio is the fixed number you multiply each term by to get to the next term.

For example, in a geometric sequence such as 2, 6, 18,..., each term is obtained by multiplying the previous term by 3. Here, the common ratio is 3. This consistency in multiplication is what classifies a sequence as geometric.

• Calculate the ratio of the second term divided by the first term.
• Continue this division all through the terms you have."
• If these calculated ratios are all equivalent, then this identical number is your common ratio."

In the exercise given, because the calculated ratios vary (4.75, 1.79, and 1.44), this sequence is not geometric and doesn't have a common ratio.

Consecutive Terms

Consecutive terms in a sequence are simply the terms that appear one after another. In the exercise's sequence, consecutive terms are like this: 4 follows none, 19 follows 4, 34 follows 19, and 49 follows 34. Think of consecutive terms as stepping stones.

In a sequence, not every set of consecutive terms forms a geometric pattern. To be a geometric sequence, the ratio between consecutive terms should remain consistent. Yet, in everyday sequences, like numbers given in the exercise, consecutive terms often reveal if a sequence breaks pattern, like with non-geometric sequences, where the ratios between these terms do not match consistently.

Applying this to the exercise, because the step between each consecutive term varies, it becomes clear that not all sequences with consecutive steps maintain the same ratio and thus may not form a geometric pattern.

Non-Geometric Sequence

A non-geometric sequence is simply a sequence that does not have a consistent common ratio between consecutive terms. These sequences are more mixed, flexible, and might not follow an obvious multiplication pattern.

• The differences are calculated, but not consistently multiplied.
• The terms can seem random or follow some other kind of rule or pattern.

In our exercise, since the ratios are not the same across consecutive terms, it is identified as non-geometric.

Understanding non-geometric sequences is important because it helps delineate various kinds of number patterns we see. Many of the number patterns encountered in real world applications might not strictly tally with a geometric structure. Recognizing when a sequence doesn't fit this pattern saves time and effort in analysis.