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Chapter 14: Problem 2

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

Short Answer

Expert verified
The first five terms are 5, 6, 7, 8, and 9.

Step by step solution

01

Identify the Formula

The given sequence is defined by the formula: a_{n}=n+4.
02

Calculate the First Term

For the first term, set n = 1 in the formula: a_{1}=1+4=5.
03

Calculate the Second Term

For the second term, set n = 2 in the formula: a_{2}=2+4=6.
04

Calculate the Third Term

For the third term, set n = 3 in the formula: a_{3}=3+4=7.
05

Calculate the Fourth Term

For the fourth term, set n = 4 in the formula: a_{4}=4+4=8.
06

Calculate the Fifth Term

For the fifth term, set n = 5 in the formula: a_{5}=5+4=9.

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Key Concepts

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

sequence formula
A sequence formula is a mathematical expression that defines the rule for generating each term in a sequence. In an arithmetic sequence, the formula usually involves an initial term and a common difference. For this exercise, we have the formula:

\( a_{n} = n + 4 \).

This formula means that each term of the sequence is generated by adding 4 to the term's position number (n).

For example:
  • If n = 1, then the term is 1 + 4 = 5.
  • If n = 2, then the term is 2 + 4 = 6.
This straightforward formula makes it easy to calculate any term in the sequence.
term calculation
Term calculation involves substituting a specific value of n into the sequence formula to find a particular term's value. Let's illustrate term calculation with the given sequence formula:

\(a_{n} = n + 4\).

Here is a step-by-step approach:

  • First term: Set \( n = 1\), then \( a_{1} = 1 + 4 = 5 \).
  • Second term: Set \( n = 2\), then \( a_{2} = 2 + 4 = 6 \).
  • Third term: Set \( n = 3\), then \( a_{3} = 3 + 4 = 7 \).
  • Fourth term: Set \( n = 4\), then \( a_{4} = 4 + 4 = 8 \).
  • Fifth term: Set \( n = 5\), then \( a_{5} = 5 + 4 = 9 \).
Calculating each term like this helps ensure we follow the rule set by the sequence formula.
arithmetic progression
An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the 'common difference.' For the current exercise, the sequence provided is:

\( a_{n} = n + 4 \).

To identify the common difference, we can calculate the difference between consecutive terms:
  • Second term - First term = 6 - 5 = 1.
  • Third term - Second term = 7 - 6 = 1.
  • Fourth term - Third term = 8 - 7 = 1.
  • Fifth term - Fourth term = 9 - 8 = 1.
In this sequence, the common difference is consistently 1. Therefore, the arithmetic progression rule states:

Next term = Previous term + Common difference.

This rule helps to extend the sequence further if needed.

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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. $$ 1,2,3,4,5, \ldots $$

Write out the first five terms of each sequence. $$ a_{n}=n+1 $$

Explain the basic difference between a sequence and a series.

Write out each series and evaluate it. $$ \sum_{i=1}^{6}(i+9) $$

Find the indicated term for each sequence. $$ a_{n}=(n+1)(2 n+3) ; \quad a_{8} $$
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