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Chapter 11: Problem 31

Predict the general term, or nth term, \(a_{m}\) of the sequence. Answers may vary. $$0, \log 10, \log 100, \log 1000, \dots$$

Short Answer

Expert verified
The nth term is \(a_{n} = n \log 10\).

Step by step solution

01

- Identify the Sequence Pattern

Examine each term of the sequence: \(0, \log 10, \log 100, \log 1000, \dots\). Notice that each term appears to be a logarithm of a power of 10.
02

- Express Each Term in Logarithmic Form

Rewrite each term in the sequence as follows: \(0 = \log 1, \ \log 10 = \log 10^1, \ \log 100 = 2 \log 10, \ \log 1000 = 3 \log 10\).
03

- General Form Identification

Recognize the pattern: each term can be written as \(n \log 10\). Where \(n\) is the position of the term in the sequence, starting from 0.
04

- Write the General Term

The general term, or nth term, \(a_{n}\) of the sequence can be expressed as: \(a_{n} = n \log 10\).

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Key Concepts

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

Logarithms
Let's start with understanding what logarithms are. Logarithms are the inverse operation to exponentiation. In simpler terms, they tell us what exponent we need to get a specific number. For example, if we have \[ \text{log}_{10}(100) = 2 \] this means that 10 raised to the power of 2 equals 100.
Logarithms help us deal with very large numbers by breaking them down into their exponential components.
Term Position
Term position refers to where a number is located within a sequence. In our exercise, we explored a sequence: \[ 0, \text{log} 10, \text{log} 100, \text{log} 1000, \text{...} \] To make predictions about the sequence, we need to clearly know each term's position.
For example:
  • The first term in the sequence (position 0) is 0, rewritten as \text{log} 1.
  • The second term (position 1) is \text{log} 10.
  • The third term (position 2) is \text{log} 100, which also is 2 \text{log} 10.
Knowing the term position helps identify patterns and understand the sequence better.
Logarithmic Pattern
A logarithmic pattern is a sequence where each term is derived using a logarithm. In our example, the sequence follows a noticeable pattern based on the powers of 10.
We write these terms as:
  • \[ \text{log} 1 = 0 \]
  • \[ \text{log} 10 = \text{log}10^1 = 1 \text{log} 10 \]
  • \[ \text{log} 100 = \text{log}10^2 = 2 \text{log} 10 \]
  • \[ \text{log} 1000 = \text{log}10^3 = 3 \text{log} 10 \]
Recognizing this pattern allows us to derive the general term, or nth term, of the sequence, which can be expressed as: \[ a_{n} = n \text{log} 10 \] This general term simplifies the process of finding any term in the sequence effortlessly.

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