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Chapter 14: Problem 8

Write the first five terms of each sequence. $$ a_{n}=n+4 $$

Short Answer

Expert verified
5, 6, 7, 8, 9

Step by step solution

01

Identify the sequence formula

Recognize that the given formula for the sequence is \[a_{n} = n + 4\].This formula is used to generate the terms of the sequence by substituting integer values for \(n\).
02

Substitute the first term

To find the first term, substitute \(n = 1\) into the formula: \[a_{1} = 1 + 4 = 5\]So, the first term is 5.
03

Substitute the second term

To find the second term, substitute \(n = 2\) into the formula: \[a_{2} = 2 + 4 = 6\]So, the second term is 6.
04

Substitute the third term

To find the third term, substitute \(n = 3\) into the formula: \[a_{3} = 3 + 4 = 7\]So, the third term is 7.
05

Substitute the fourth term

To find the fourth term, substitute \(n = 4\) into the formula: \[a_{4} = 4 + 4 = 8\]So, the fourth term is 8.
06

Substitute the fifth term

To find the fifth term, substitute \(n = 5\) into the formula: \[a_{5} = 5 + 4 = 9\]So, the fifth term is 9.

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Key Concepts

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

Understanding Algebraic Sequences
Algebraic sequences are a type of sequence where each term is created by performing algebraic operations on a variable, typically represented by n. In our example, we have the sequence formula \(a_{n} = n + 4\). To generate each term in the sequence, we simply substitute different integer values for n and solve the resulting expression.The process is straightforward:
  • Identify the formula used to create the sequence.
  • Substitute integer values into the formula step by step.
  • Calculate the value of each term.
In our case, adding 4 to successive integer values of n generates each term of the sequence. Sequences like this are fundamental in algebra and form the basis for understanding more complex mathematical concepts.
Using Integer Substitution
Integer substitution is the technique of replacing the variable n with specific integer values to generate individual terms in a sequence. This technique is essential for solving various types of algebraic sequences.Let's illustrate this using our sequence formula \(a_{n} = n + 4 \):
  • Start with the first integer, n = 1: \[a_1 = 1 + 4 = 5\]
  • Next, substitute n = 2: \[a_2 = 2 + 4 = 6\]
  • Continue with n = 3: \[a_3 = 3 + 4 = 7\]
  • Next, use n = 4: \[a_4 = 4 + 4 = 8\]
  • Finally, substitute n = 5: \[a_5 = 5 + 4 = 9\]
Each step involves straightforward arithmetic calculations, demonstrating how integer substitution helps in understanding and generating terms of the sequence.
Term Generation in Sequences
Term generation is the method of calculating each term in a sequence from a given formula. In our algebraic sequence example, each term is generated by applying the sequence formula \(a_{n} = n + 4\) to various integer values of n.Here's a recap of the term generation process for the first five terms:
  • First term: \[a_1 = 1 + 4 = 5\]
  • Second term: \[a_2 = 2 + 4 = 6\]
  • Third term: \[a_3 = 3 + 4 = 7\]
  • Fourth term: \[a_4 = 4 + 4 = 8\]
  • Fifth term: \[a_5 = 5 + 4 = 9\]
By following these steps, students can understand the importance of the formula and the easy substitution process involved in generating each term in an algebraic sequence.

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Most popular questions from this chapter

If the given sequence is arithmetic, find the common difference \(d .\) If the sequence is not arithmetic, say so. See Example 1. \(2,5,8,11, \ldots\)

Find the indicated term for each sequence. $$ a_{n}=-9 n+2 ; \quad a_{8} $$

Determine an expression for the general term of each arithmetic sequence. Then find \(a_{25}\). \(1, \frac{5}{3}, \frac{7}{3}, 3, \ldots\)

Fill in each blank with the correct response. In a geometric sequence, if any term after the first is divided by the term that precedes it, the result is the common _______ of the sequence.

Determine an expression for the general term \(a_{n}\) of each sequence \(-10,-20,-30,-40, \ldots\)
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