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Chapter 14: Problem 19

Write the first five terms of each sequence and then find the specified term. $$ a_{n}=-3 n+1, a_{30} $$

Short Answer

Expert verified
The first five terms are -2, -5, -8, -11, -14, and the 30th term is -89.

Step by step solution

01

Understanding the sequence formula

The given sequence formula is \( a_n = -3n + 1 \). This means that each term of the sequence is calculated by multiplying the term number \( n \) by -3 and then adding 1.
02

Calculate the first term

To find the first term (\( a_1 \)), substitute \( n = 1 \) into the formula: \( a_1 = -3(1) + 1 = -3 + 1 = -2 \). So, the first term is -2.
03

Calculate the second term

Substitute \( n = 2 \) into the formula to find the second term: \( a_2 = -3(2) + 1 = -6 + 1 = -5 \). So, the second term is -5.
04

Calculate the third term

Substitute \( n = 3 \) into the formula to find the third term: \( a_3 = -3(3) + 1 = -9 + 1 = -8 \). So, the third term is -8.
05

Calculate the fourth term

Substitute \( n = 4 \) into the formula to find the fourth term: \( a_4 = -3(4) + 1 = -12 + 1 = -11 \). So, the fourth term is -11.
06

Calculate the fifth term

Substitute \( n = 5 \) into the formula to find the fifth term: \( a_5 = -3(5) + 1 = -15 + 1 = -14 \). So, the fifth term is -14.
07

Calculate the 30th term

To find the 30th term, substitute \( n = 30 \) into the formula: \( a_{30} = -3(30) + 1 = -90 + 1 = -89 \). So, the 30th term is -89.

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Key Concepts

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

Understanding the Sequence Formula
A sequence formula provides a way to generate the terms of a sequence. In the problem here, we have the sequence formula:
  • \( a_n = -3n + 1 \)
This means each term in the sequence can be calculated using this equation, where \( n \) represents the position of the term in the sequence. By plugging different integer values of \( n \) into the formula, you can quickly calculate the corresponding terms.
For example:
  • If \( n = 1 \), then \( a_1 = -3 \times 1 + 1 \), which simplifies to -2.
  • If \( n = 2 \), then \( a_2 = -3 \times 2 + 1 \), resulting in -5.
  • Continue substituting to find other terms easily.
This straightforward approach makes the sequence formula a powerful tool in arithmetic sequences, allowing us to derive any term of the sequence without the need to calculate all preceding terms.
Calculating Terms in a Sequence
Calculating terms in an arithmetic sequence involves substituting different values of \( n \) into the sequence formula. For the given sequence:
  • Start with \( n = 1 \) to find the first term.
  • Increase \( n \) by 1 each time to find subsequent terms.
For instance, in the sequence given:
  • The first term, \( a_1 \), is found by calculating \( a_1 = -3(1) + 1 = -2 \).
  • The second term, \( a_2 \), using \( n = 2 \) gives \( a_2 = -3(2) + 1 = -5 \).
  • Continue this pattern to find more terms such as the third term \( a_3 = -8 \), fourth term \( a_4 = -11 \), and fifth term \( a_5 = -14 \).
This method allows us to quickly determine any term in the sequence, including terms much further along in the sequence, such as the 30th term, \( a_{30} = -89 \). This seamless calculation process makes focusing on specific terms much easier.
Role of Algebraic Expressions
Algebraic expressions play a crucial role in sequences as they enable us to represent sequences in a general form. By using algebra, sequences can be defined and understood clearly. Let's break down how algebraic expressions work in this context:
  • The sequence formula \( a_n = -3n + 1 \) is itself an algebraic expression.
  • This expression establishes a relationship between a term number \( n \) and the actual term value \( a_n \).
  • Algebraic expressions like this offer a clear, concise way to express complex numerical relationships.
Algebraic notation simplifies the process of term calculation and helps in seeing the broader pattern or relationship within a sequence. This, in turn, facilitates better understanding and manipulation of sequences in algebra and arithmetic.

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