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Magazine Chapter 9: Problem 46
For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{100 \cdot n}{n(n-1) !} $$
Short Answer
Expert verified
The first four terms are: 100, 100, 50, and approximately 16.67.
Step by step solution
01
Understand the Given Sequence Formula
The sequence is defined by the formula \( a_{n} = \frac{100 \cdot n}{n(n-1)!} \). This formula describes how to calculate each term \( a_{n} \) in the sequence depending on the index \( n \). "!" denotes the factorial operation.
02
Calculate the First Term (n=1)
Substitute \( n = 1 \) into the sequence formula: \[a_{1} = \frac{100 \cdot 1}{1(1-1)!} = \frac{100}{1 \cdot 0!} = \frac{100}{1 \cdot 1} = 100\]
03
Calculate the Second Term (n=2)
Substitute \( n = 2 \) into the sequence formula: \[a_{2} = \frac{100 \cdot 2}{2(2-1)!} = \frac{200}{2 \cdot 1!} = \frac{200}{2 \cdot 1} = 100\]
04
Calculate the Third Term (n=3)
Substitute \( n = 3 \) into the sequence formula: \[a_{3} = \frac{100 \cdot 3}{3(3-1)!} = \frac{300}{3 \cdot 2!} = \frac{300}{3 \cdot 2} = 50\]
05
Calculate the Fourth Term (n=4)
Substitute \( n = 4 \) into the sequence formula: \[a_{4} = \frac{100 \cdot 4}{4(4-1)!} = \frac{400}{4 \cdot 3!} = \frac{400}{4 \cdot 6} = \frac{400}{24} \approx 16.67\]
06
Summarize the First Four Terms
The first four terms of the sequence, based on the calculations, are as follows:- \( a_{1} = 100 \)- \( a_{2} = 100 \)- \( a_{3} = 50 \)- \( a_{4} \approx 16.67 \)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factorial Operation
The factorial operation is a fundamental mathematical concept signified by an exclamation mark "!". When you see an expression like "n!", it means you're looking at the product of all positive integers from 1 up to n. For instance, if you see "5!", you're calculating \(5 \times 4 \times 3 \times 2 \times 1\), which equals 120.
Factorials are pivotal in various mathematical fields, especially in sequences and permutations. A few key points about factorial operation:
Factorials are pivotal in various mathematical fields, especially in sequences and permutations. A few key points about factorial operation:
- "0!" is a special case and is defined as 1.
- Factorials grow very quickly, which affects calculations in mathematics and computer science.
- "n!" is only defined for non-negative integers.
Term of a Sequence
In any sequence, a 'term' refers to each individual element within that sequence. Sequences are typically defined by mathematical formulas that determine what each term is based on its position, known as its index.
For example, in the sequence given by the formula \( a_{n} = \frac{100 \cdot n}{n(n-1)!} \), each term is calculated by substituting different integer values for n. Each term gives a specific output depending on this formula, creating a list of numbers when computed for consecutive integers.Some considerations for understanding sequence terms:
For example, in the sequence given by the formula \( a_{n} = \frac{100 \cdot n}{n(n-1)!} \), each term is calculated by substituting different integer values for n. Each term gives a specific output depending on this formula, creating a list of numbers when computed for consecutive integers.Some considerations for understanding sequence terms:
- Sequences can be finite or infinite.
- Each term's value depends on its position in the sequence outlined by its formula.
- The first term is often what you get when substituting n with 1, the second with 2, and so on.
Index in Sequences
The 'index' refers to the position of a term within a sequence. It is denoted by "n" in mathematical formulas representing sequences. The index is crucial because it indicates which term we're referring to or calculating.
In our sequence \( a_{n} = \frac{100 \cdot n}{n(n-1)!} \), the index "n" signifies the specific term we are calculating in the sequence. For example, when n equals 3, we calculate the third term of the sequence.Important properties of indices include:
In our sequence \( a_{n} = \frac{100 \cdot n}{n(n-1)!} \), the index "n" signifies the specific term we are calculating in the sequence. For example, when n equals 3, we calculate the third term of the sequence.Important properties of indices include:
- Indices start from 1 or 0, depending on the sequence context.
- They're integer values that increase as you move through the sequence.
- The index determines the application of the formula to obtain a term.
Sequence Calculation Steps
Calculating terms in a sequence involves substituting values for the index into the sequence's formula and solving for each. The goal is to find the value of each term, like the terms of our sequence provided by \( a_{n} = \frac{100 \cdot n}{n(n-1)!} \). Necessary steps generally follow this pattern:
- Identify the formula and understand each component: Know what each part of the formula does. Here, "n" is the index, "100" is a constant, and "(n-1)!" represents a factorial operation.
- Select the index value: Decide which term you are calculating. For the first term, use n=1, second term n=2, etc.
- Substitute the index value into the formula: Replace "n" with your selected index number and solve the arithmetic operations.
- Compute the factorial (if applicable): Calculate the factorial of the applicable numbers. For example, "2!" is 2.
- Perform final arithmetic operations: Multiply, divide, or simplify as required to arrive at the term's value.
