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Chapter 10: Problem 12

Write the first six terms of each arithmetic sequence. $$a_{n}=a_{n-1}-20, a_{1}=50$$

Short Answer

Expert verified
The function generates the first six terms of the arithmetic sequence as: \(50, 30, 10, -10, -30, -50\).

Step by step solution

01

Identify the initial term

The initial term of an arithmetic sequence, labeled as \(a_1\), is given in the problem as 50. Hence, the first term of the sequence is 50.
02

Apply the given arithmetic sequence formula to find the next terms

The formula given is \(a_n= a_{n-1}-20\). By applying this formula, the next term can be found by subtracting 20 from the previous term. The following terms will thus be - \n\(a_2= a_1 -20= 50 -20 = 30\)\n\(a_3= a_2 - 20 = 30 - 20 = 10\)\n\(a_4= a_3 -20 = 10 -20 = -10\)\n\(a_5= a_4 -20 = -10 -20 = -30\)\n\(a_6= a_5 -20 = -30 -20 = -50\)
03

List out the first six terms of the sequence

The first six terms of the given arithmetic sequence are: 50, 30, 10, -10, -30, -50.

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Key Concepts

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

Arithmetic Sequence Formula
When we talk about an arithmetic sequence, we are referring to a series of numbers where each term after the first is found by adding a constant, known as the common difference, to the previous term. The general arithmetic sequence formula is
\[\begin{equation} a_n = a_1 + (n - 1)d \end{equation}\]
In this formula,
  • a_n represents the nth term of the sequence,
  • a_1 is the first term,
  • d is the common difference, and
  • n signifies the position of the term in the sequence.

It's essential to understand this formula because it allows you to calculate any term within the sequence without knowing all the previous terms. This is particularly useful when dealing with large sequences, where calculating each term individually would be impractical.
Initial Term of an Arithmetic Sequence
The starting point of an arithmetic sequence is known as the initial term, or a_1. Knowing the initial term is crucial, as it sets the stage for the entire sequence. Think of it as the foundation upon which the sequence is built. In the given exercise, the initial term is told to be 50. This value is essential as without knowing the first term, you cannot effectively utilize the arithmetic sequence formula to find subsequent terms. When crafting your sequence, always ensure you have the initial term at hand, as it is the first building block of your arithmetic structure.
Finding Terms of an Arithmetic Sequence
Once you have the initial term and the common difference, finding other terms in an arithmetic sequence is a straightforward process. By repeatedly adding the common difference, you can quickly determine any term in the sequence. In the given problem, the common difference is -20 since we subtract 20 to find the next term. Let's put it into practice:
  • The second term (a_2) is the first term (a_1 = 50) plus one common difference (d = -20), which equals 30.
  • Similarly, for the third term (a_3), you add the common difference to the second term, giving you 10.

Repeat this process for the remaining terms, and you will find the sequence: 50, 30, 10, -10, -30, -50. By mastering the method of finding terms in an arithmetic sequence, you gain the ability to analyze and determine the nature of sequences, predict future terms, and understand the underlying patterns within sequences.

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Most popular questions from this chapter

Mega Millions is a multi-state lottery played in most U.S. states. As of this writing, the top cash prize was \(\$ 656\) million, going to three lucky winners in three states. Players pick five different numbers from 1 to 56 and one number from 1 to \(46 .\) Use this information to solve Exercises \(27-30 .\) Express all probabilities as fractions. A player wins the jackpot by matching all five numbers drawn from white balls ( 1 through 56 ) and matching the number on the gold Mega Ball \(^{\oplus}\) ( 1 through 46 ). What is the probability of winning the jackpot?

Show that \(B\) is the multiplicative inverse of \(A,\) where $$ A=\left[\begin{array}{ll} 2 & 3 \\ 1 & 2 \end{array}\right] \text { and } B=\left[\begin{array}{rr} 2 & -3 \\ -1 & 2 \end{array}\right] $$

Solve triangle \(A B C\) if \(a=17, b=28,\) and \(c=15\) Round angle measures to the nearest degree. (Section 6.2, Example 2) (GRAPH CANT COPY)

This will help you prepare for the material covered in the next section. Consider the sequence \(1,-2,4,-8,16, \ldots\) Find \(\frac{a_{2}}{a_{1}}, \frac{a_{3}}{a_{2}}, \frac{a_{4}}{a_{3}}\) and \(\frac{a_{5}}{a_{4}} .\) What do you observe?

a. If two people are selected at random, the probability that they do not have the same birthday (day and month) is \(\frac{365}{365} \cdot \frac{364}{365}\). Explain why this is so. (Ignore leap years and assume 365 days in a year.) b. If three people are selected at random, find the probability that they all have different birthdays. c. If three people are selected at random, find the probability that at least two of them have the same birthday. d. If 20 people are selected at random, find the probability that at least 2 of them have the same birthday. e. How large a group is needed to give a 0.5 chance of at least two people having the same birthday?
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