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Magazine Chapter 14: Problem 9
Write the first five terms of each sequence. $$ a_{n}=\frac{n+3}{n} $$
Short Answer
Expert verified
The first five terms are: 4, \( \frac{5}{2} \), 2, \( \frac{7}{4} \), and \( \frac{8}{5} \).
Step by step solution
01
Understand the Function
The sequence is given by the function \( a_{n} = \frac{n+3}{n} \). For each term in the sequence, substitute \( n \) with the term number (starting from 1) into this function.
02
Find the First Term
To find the first term, substitute \( n = 1 \) into the function: \( a_{1} = \frac{1+3}{1} = \frac{4}{1} = 4 \). So, the first term is 4.
03
Find the Second Term
To find the second term, substitute \( n = 2 \) into the function: \( a_{2} = \frac{2+3}{2} = \frac{5}{2} \). So, the second term is \( \frac{5}{2} \).
04
Find the Third Term
To find the third term, substitute \( n = 3 \) into the function: \( a_{3} = \frac{3+3}{3} = \frac{6}{3} = 2 \). So, the third term is 2.
05
Find the Fourth Term
To find the fourth term, substitute \( n = 4 \) into the function: \( a_{4} = \frac{4+3}{4} = \frac{7}{4} \). So, the fourth term is \( \frac{7}{4} \).
06
Find the Fifth Term
To find the fifth term, substitute \( n = 5 \) into the function: \( a_{5} = \frac{5+3}{5} = \frac{8}{5} \). So, the fifth term is \( \frac{8}{5} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
sequence
A sequence is a list of numbers that follow a specific pattern or rule. In mathematics, sequences highlight ordered sets that often depend on a variable or index. Consider the sequence \(\frac{n+3}{n}\) which generates terms based on the input values substituted into it.
The key points to understand sequences include:
The key points to understand sequences include:
- Each term in a sequence is generated based on a specific formula or rule.
- The index typically starts from 1 or sometimes 0, depending on the context or definition.
- Sequences are often represented in the form of \(a_n\) where \(n\) determines the position in the sequence.
substitution
Substitution involves replacing a variable with a particular value to solve an equation or derive a term in a sequence. It's a fundamental technique in mathematics used to evaluate expressions.
For example:
For example:
- Substituting \(n = 1\) means replacing \(n\) with 1 in the function. For \(a_n = \frac{n+3}{n}\), when \(n = 1\), it becomes \(a_{1} = \frac{1+3}{1} = 4\).
- To find the second term, substitute \(n = 2\), resulting in \(a_{2}= \frac{2+3}{2}= \frac{5}{2} \).
term calculation
Term calculation refers to finding specific values in a sequence using its rule. For instance, in the sequence \(a_n = \frac{n+3}{n}\), each term of the sequence is derived by substituting the appropriate number for \(n\). Here is the breakdown for the first five terms:
- 1st term: \(a_{1} = \frac{1+3}{1} = 4 \)
- 2nd term: \(a_{2}= \frac{2+3}{2}= \frac{5}{2} \)
- 3rd term: \(a_{3}= \frac{3+3}{3}= 2\)
- 4th term: \(a_{4}= \frac{4+3}{4}= \frac{7}{4} \)
- 5th term: \(a_{5}= \frac{5+3}{5}= \frac{8}{5} \)
rational functions
Rational functions are mathematical expressions represented as the ratio of two polynomials. Such functions showcase relationships that can be applied to sequences and term calculations.
In a general form, a rational function is:
\ f(x) = \frac{P(x)}{Q(x)} \
Where \(P(x) \) and \(Q(x) \) are polynomials, and \(Q(x) \) is not zero. Analyzing the sequence \(a_n = \frac{n+3}{n}\) reveals it's a rational function where:
In a general form, a rational function is:
\ f(x) = \frac{P(x)}{Q(x)} \
Where \(P(x) \) and \(Q(x) \) are polynomials, and \(Q(x) \) is not zero. Analyzing the sequence \(a_n = \frac{n+3}{n}\) reveals it's a rational function where:
- \( P(n) = n + 3 \)
- \(Q(n) = n \)
