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Chapter 12: Problem 2

Write out the first five terms of each sequence. $$ \text { 2. } a_{n}=n+4 $$

Short Answer

Expert verified
5, 6, 7, 8, 9

Step by step solution

01

Identify the sequence formula

The given sequence formula is \(a_n = n + 4\). This formula will be used to calculate each term in the sequence.
02

Calculate the first term

Substitute \(n = 1\) into the formula: \(a_1 = 1 + 4 = 5\). Therefore, the first term is 5.
03

Calculate the second term

Substitute \(n = 2\) into the formula: \(a_2 = 2 + 4 = 6\). Therefore, the second term is 6.
04

Calculate the third term

Substitute \(n = 3\) into the formula: \(a_3 = 3 + 4 = 7\). Therefore, the third term is 7.
05

Calculate the fourth term

Substitute \(n = 4\) into the formula: \(a_4 = 4 + 4 = 8\). Therefore, the fourth term is 8.
06

Calculate the fifth term

Substitute \(n = 5\) into the formula: \(a_5 = 5 + 4 = 9\). Therefore, 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.

Arithmetic Sequence
An **arithmetic sequence** is a type of number sequence where the difference between consecutive terms is always the same. This difference is called the common difference. For instance, in the sequence 5, 6, 7, 8, 9, the common difference is 1.
To recognize an arithmetic sequence, you can check if subtracting any term from the next one gives you a constant value.
Here, the exercise gives us a sequence formula: \(a_n = n + 4\). The pattern shows that every next term is consistently greater than the previous one by 1. This confirms it is an arithmetic sequence.
Term Calculation
The process of calculating individual terms in a sequence is vital to understanding sequences. Using a given sequence formula helps you to find specific terms by substituting different values of n.
For example, our formula is \(a_n = n + 4\). To find the first term \(a_1\), you substitute n with 1:
  • \(a_1 = 1 + 4 = 5\)
To find the second term \(a_2\), substitute n with 2:
  • \(a_2 = 2 + 4 = 6\)
This method extends to any term you want to find. Each term is simply derived by substituting the position number into the sequence formula.
Algebraic Expression in Sequences
An **algebraic expression** in sequences is a mathematical statement involving numbers, variables, and operational symbols that defines the pattern of the sequence.
In our case, the algebraic expression is \(a_n = n + 4\). It specifies how each term in the sequence is generated.
Here, 'n' represents the position of a term in the sequence. For example:
  • When n = 1: \(a_1 = 1 + 4 = 5\)
  • When n = 2: \(a_2 = 2 + 4 = 6\)
This algebraic expression helps to derive any term in the sequence without needing to list all preceding terms.

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Most popular questions from this chapter

Find the arithmetic mean for each collection of numbers. $$ 10,12,8,19,23,12 $$

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