Problem 12 - 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. $$9,-6,4,-\frac{8}{3}, \dots$$

Step-by-Step Solution

Answer

Yes, the sequence is geometric and the common ratio is \( r = -2/3 \)

Step 1: Calculate the ratio between consecutive terms

The ratio (\(r\)) between two consecutive terms in a geometric sequence is given by \( r = \frac{T_{n+1}}{T_{n}} \), where \(T_{n}\) represents the \(n\)th term in the sequence. So, calculate the ratio between the first and second term, then calculate the ratio between the second and third term. In our case, \( r_{1,2} = \frac{-6}{9} = -2/3, r_{2,3} = \frac{4}{-6} = -2/3 \)

Step 2: Check if the ratio is consistent

If the sequence is geometric, the ratio must be the same for all pairs of consecutive terms. The ratios calculated in Step 1 are equal, so now check the ratio between the third and fourth terms. Our ratio is \( r_{3,4} = \frac{-8/3}{4} = -2/3 \), the same as before. Therefore, the sequence is geometric.

Step 3: Identify the common ratio

If a sequence is geometric and the ratio between each pair of consecutive terms is consistent, that ratio is the common ratio. In our case, the common ratio is \( r = -2/3 \).

Key Concepts

Common Ratio

In any geometric sequence, the common ratio is a crucial concept. This ratio is what allows a sequence to maintain its unique geometric characteristics. To put it simply, the common ratio is the constant factor by which each term is multiplied to get the next term.
To find the common ratio, divide any term in the sequence (after the first) by the term that immediately precedes it. Mathematically, it is expressed as \( r = \frac{T_{n+1}}{T_{n}} \), where \( T_{n} \) is a term in the sequence and \( T_{n+1} \) is the subsequent term.
Why is this important? Understanding this ratio helps us determine how the sequence progresses and ensures its geometric nature. In the sequence \(9, -6, 4, -\frac{8}{3}, \, \dots \), the common ratio \( r \) is \(-\frac{2}{3}\).
Every time one of these terms is multiplied by \(-\frac{2}{3}\), it results in the following term.

Consecutive Terms

Consecutive terms in a sequence are simply terms that follow one another in turn. In a geometric sequence, consecutive terms help us to determine the common ratio quickly and directly.
When we take two terms that are next to each other in the sequence — let's call them \( T_{n} \) and \( T_{n+1} \) — we can calculate the ratio between them. If this computation results in the same value - the common ratio - across all pairs of consecutive terms, the sequence is confirmed to be geometric.
In our example sequence \(9, -6, 4, -\frac{8}{3}, \, \dots \), observe how each term is related to the one before it. By finding the ratio \( \frac{-6}{9} = -\frac{2}{3} \) and knowing it holds for other consecutive pairs, we confirm the nature of the sequence.

Consistent Ratio

While studying geometric sequences, one key characteristic to focus on is the consistent ratio. A sequence is only considered geometric if this ratio remains the same across all pairs of consecutive terms throughout the sequence, indicating a linear multiplication factor.
For instance, if one term divided by the previous term equals the same value for every consecutive pair, you can confidently state that the sequence is geometric. This consistent ratio is a fail-safe test for the sequence's geometric property.
For the sequence \(9, -6, 4, -\frac{8}{3}, \, \dots \), since the ratio between each pair of consecutive terms is consistently \(-\frac{2}{3}\), the sequence can be confirmed as geometric. This aspect is central in classifying and understanding geometric sequences.