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Chapter 19: Problem 29

Find the indicated quantities.Is \(3,3^{x+1}, 3^{2 x+1}, \ldots\) a geometric sequence? Explain. If it is, find \(a_{20}\).

Short Answer

Expert verified
Yes, it is a geometric sequence, and \(a_{20} = 3^{19x+1}\).

Step by step solution

01

Determine if the sequence is geometric

A sequence is geometric if there is a constant ratio, called the common ratio, between consecutive terms. The given sequence is \(3, 3^{x+1}, 3^{2x+1}, \ldots\). To find the common ratio, compute the ratios of successive terms. The ratio of the second term to the first term is \(\frac{3^{x+1}}{3} = 3^x\). The ratio of the third term to the second term is \(\frac{3^{2x+1}}{3^{x+1}} = 3^x\). Since both ratios are the same, the sequence is geometric with a common ratio of \(3^x\).
02

Identify the first term of the sequence

In any geometric sequence, the first term \(a_1\) is the starting number of the sequence. Here, the first term \(a_1\) is given as 3.
03

Use the formula to find the nth term of a geometric sequence

The general formula for the nth term of a geometric sequence is \(a_n = a_1 \cdot r^{(n-1)}\), where \(a_1\) is the first term and \(r\) is the common ratio. For this sequence, \(a_1 = 3\) and \(r = 3^x\). Substitute these into the formula to get \(a_n = 3 \cdot (3^x)^{(n-1)}\).
04

Calculate the 20th term \(a_{20}\)

Now substitute \(n = 20\) into the nth term formula: \(a_{20} = 3 \cdot (3^x)^{19}\). This simplifies to \(a_{20} = 3 \cdot 3^{19x} = 3^{1 + 19x}\). Therefore, \(a_{20} = 3^{19x+1}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Ratio
In a geometric sequence, one of the vital concepts is the **common ratio**. This is the ratio between any two consecutive terms in the sequence. To determine if a sequence is geometric, check if this ratio is consistent, or remains the same, for all term pairs.
  • For instance, in the sequence given in the exercise, the first few terms are \(3, 3^{x+1}, 3^{2x+1}, \ldots\)
  • We calculate the ratio between the second term and the first term: \(\frac{3^{x+1}}{3} = 3^x\).
  • Again, between the third term and the second term: \(\frac{3^{2x+1}}{3^{x+1}} = 3^x\).
This consistent ratio, \(3^x\), confirms the sequence is geometric. The concept of the common ratio is essential as it confirms the uniformity and structure of a geometric sequence.
Nth Term Formula
The nth term formula is a tool that allows us to find any term in a geometric sequence, without listing all the terms. This formula is expressed as:
  • \(a_n = a_1 \cdot r^{(n-1)}\) where:
  • \(a_1\) is the first term of the sequence,
  • \(r\) is the common ratio, and
  • \(n\) is the position of the term you want to find.
For the provided sequence:
  • The first term, \(a_1\), is 3.
  • The common ratio, \(r\), is \(3^x\).
  • Thus, the nth term formula becomes \(a_n = 3 \cdot (3^x)^{n-1}\).
Understanding this formula makes it easy to calculate any term quickly, as we did for the 20th term in the exercise.
Sequence Analysis
**Sequence analysis** involves examining the pattern and structure of a sequence. Here, it involves confirming that repetitive patterns exist and comply with the rules of geometric sequences.
  • Initially, we identify the common ratio among the terms, ensuring a consistent mathematical pattern.
  • Then, we utilize the nth term formula to not only confirm the pattern but also to predict future terms.
  • This step remains crucial for solidifying the understanding of how terms evolve within the sequence.
By thoroughly analyzing each component, students reinforce their grasp of key concepts, enabling them to tackle even more complex sequence-related exercises confidently.
Exponential Sequences
Geometric sequences like the one in this exercise are often examples of **exponential sequences** due to the exponential nature in which they expand. Let's explore why:
  • In a geometric sequence, the ratio between terms is constantly multiplied by the common ratio, which is an exponential factor.
  • This repeated multiplication leads to the sequence growing at an exponential rate.
  • For instance, each term in the sequence \(3^{x+1}, 3^{2x+1}, \ldots\) involves raising to increasing powers, further highlighting their exponential properties.
Understanding geometric sequences as exponential sequences helps underline the rapid growth and scale of such sequences, providing a deeper insight into their nature and application.

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Most popular questions from this chapter

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