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Chapter 6: Problem 58

Find the \(n\) th term of the geometric sequence. $$0.234,0.000234,0.000000234, \ldots$$

Short Answer

Expert verified
The nth term of the geometric sequence is given by \(a_n = 0.234 \cdot (0.001)^{n-1}\).

Step by step solution

01

Determine the first term and common ratio

From the sequence, it can be determined that the first term, \(a\), is 0.234. The common ratio, \(r\), can be found by taking the second term divided by the first term, which equals 0.000234 / 0.234 = 0.001.
02

Apply the formula

The formula for the nth term of a geometric sequence is given by \(a_n = a \cdot r^{(n-1)}\). Now substitute the values of \(a\) and \(r\) into the formula.
03

Final expression for the nth term

The general form of the nth term of the sequence is thereby expressed as \(a_n = 0.234 \cdot (0.001)^{n-1}\).

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Key Concepts

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

Understanding the First Term in a Geometric Sequence
When studying geometric sequences, the **first term** is a fundamental component. In any geometric sequence, the first term is the starting point from which the rest of the sequence is generated. It's represented by the symbol \(a\). In this case, the first term of the sequence is \(0.234\).

Think of the first term as the foundation. It's the value you build upon as you progress through the sequence. Without the first term, the sequence cannot continue. Hence, identifying the first term correctly is critical before attempting to find any subsequent terms.
  • Symbol for the first term: \(a\)
  • In our sequence: \(a = 0.234\)
By understanding the first term, you set the stage for exploring the rest of the geometric sequence.
Demystifying the Common Ratio
The **common ratio** of a geometric sequence determines the factor by which each term is multiplied to get the next term. Represented by the symbol \(r\), it is what makes the sequence 'geometric'. To find \(r\), you divide any term in the sequence by the previous term. This ratio remains constant throughout.

In our example sequence, the second term \(0.000234\) divided by the first term \(0.234\) gives a common ratio \(r = 0.001\). This tells us that each term is 0.001 times the term before it.
  • Common ratio formula: \(r = \text{next term} / \text{previous term}\)
  • In our sequence: \(r = 0.000234 / 0.234 = 0.001\)
Understanding the common ratio is essential. It reveals how terms relate to each other, enabling a deep comprehension of how the sequence evolves.
Using the nth Term Formula for Geometric Sequences
The **nth term formula** is a powerful tool in understanding geometric sequences. With this formula, you can calculate any term within the sequence without listing all previous terms. The formula is expressed as \(a_n = a \cdot r^{(n-1)}\).

Let's break it down:
  • \(a_n\) is the term you're trying to find.
  • \(a\) is the first term.
  • \(r\) is the common ratio.
  • \(n\) is the term number in the sequence.
For our sequence, the nth term is computed as \(a_n = 0.234 \cdot (0.001)^{n-1}\). So if you wanted to find, say, the 5th term, you would substitute \(n = 5\) into the formula.
This formula is an efficient way to find terms at any position in the sequence, making it invaluable for solving sequence-related problems with ease.

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Most popular questions from this chapter

Find the \(n\)th partial sum of the arithmetic sequence. $$a_{n}=3 n+2 ; n=10$$

Consider a square with a side of length \(1 .\) Construct another square inside the first one by connecting the midpoints of the sides of the first square. What is the area of the inscribed square? Continue constructing squares in the same way. Find the area of the \(n\) th inscribed square.

Write each rational number as the quotient of two integers in simplest form. $$0 . \overline{123}$$

Use a graphing utility to graph each equation. $$r=\ln \theta$$

Write each rational number as the quotient of two integers in simplest form. $$0 . \overline{3}$$
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