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Chapter 14: Problem 15

Write the first four terms of each sequence whose general term is given. $$a_{n}=2(n+1) !$$

Short Answer

Expert verified
The first four terms of the sequence are 4, 12, 48, and 240.

Step by step solution

01

Substitute n=1

Substitute n=1 into the general term expression: \(a_{1}=2(1+1) != 2(2)! = 2*2 = 4\). So, the first term of the sequence is 4.
02

Substitute n=2

Substitute n=2 into the general term expression: \(a_{2}=2(2+1) != 2(3)! = 2*6 = 12\). So, the second term of the sequence is 12.
03

Substitute n=3

Substitute n=3 into the general term expression: \(a_{3}=2(3+1) != 2(4)! = 2*24 = 48\). So, the third term of the sequence is 48.
04

Substitute n=4

Substitute n=4 into the general term expression: \(a_{4}=2(4+1) != 2(5)! = 2*120 = 240\). So, the fourth term of the sequence is 240.

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Key Concepts

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

Factorial Notation
Factorial notation is a shorthand used in mathematics to describe the product of all positive integers up to a certain number.
  • The notation is represented by an exclamation mark (!) after a number.
  • For example, for a positive integer \( n \), \( n! \) is the product \( n \times (n-1) \times (n-2) \times \, ... \, \times 1 \).
  • This sequence of multiplications is helpful in combinatorics, algebra, and calculus.
In the given exercise, factorials are used within the general term of the sequence expression. By substituting \( n \) with integers starting from 1, we compute the factorial first and then multiply by 2 as prescribed by the sequence formula. Understanding factorials is crucial since the growth of the product can be quite steep as \( n \) increases.
Sequence Terms
A sequence in mathematics is an ordered list of numbers, where each number in the list is called a "term." Terms are generated based on rules or formulas dictated by the sequence.
  • In our exercise, the general term \( a_{n} = 2(n+1)! \) symbolizes a sequence.
  • Each term is found by plugging in a different value for \( n \), starting at 1 and increasing by one each time, into the formula.
The first four terms of this setup define a pattern based on factorial calculations. As each term is calculated, the subsequent value often grows exponentially due to the nature of factorials:- The first term \( a_1 = 4 \), calculated by substituting \( n=1 \).- The second term \( a_2 = 12 \), resulting from \( n=2 \).- The third term \( a_3 = 48 \), following the use of \( n=3 \).- The fourth term \( a_4 = 240 \), achieved by calculating with \( n=4 \).
General Term of a Sequence
The general term of a sequence is a formula that allows us to find any term in the sequence without having to write out all the preceding terms.
  • It is usually denoted with the symbol \( a_{n} \), where \( n \) represents the position of a term within the sequence.
  • The provided formula for the exercise is \( a_{n} = 2(n+1)! \).
This particular general term makes it clear that each term is derived by multiplying 2 by the factorial of \( n+1 \). Knowing this allows you to generate terms easily and predict values beyond just the first few calculated ones. For instance, if you wanted to find the tenth term, you'd simply substitute \( n=10 \) into the general term expression to find \( a_{10} = 2(11)! \), demonstrating the power and simplicity of using general terms.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A degree-day is a unit used to measure the fuel requirements of buildings. By definition, each degree that the average daily temperature is below \(65^{\circ} \mathrm{F}\) is 1 degree-day. For example, an average daily temperature of \(42^{\circ} \mathrm{F}\) constitutes 23 degree-days. If the average temperature on January 1 was \(42^{\circ} \mathrm{F}\) and fell \(2^{\circ} \mathrm{F}\) for each subsequent day up to and including January \(10,\) how many degree-days are included from January 1 to January 10?

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$\frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \frac{6}{5}, \dots$$

Expand and write the answer as a single logarithm with a coefficient of 1. $$\sum_{i=1}^{4} \log (2 i)$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. In order to expand \(\left(x^{3}-y^{4}\right)^{5},\) I find it helpful to rewrite the expression inside the parentheses as \(x^{3}+\left(-y^{4}\right)\).

What is a geometric sequence? Give an example with your explanation.
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