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Chapter 8: Problem 8

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=(-1)^{n-1}(n+1)$$

Short Answer

Expert verified
The first five terms of the sequence are 2, -3, 4, -5, and 6.

Step by step solution

01

Understand the Sequence Formula

The given sequence formula is \( a_n = (-1)^{n-1}(n+1) \). This formula generates a sequence based on the index \( n \). This formula uses the alternating signs determined by \((-1)^{n-1}\), and the term \(n+1\) changes linearly with \(n\).
02

Find the First Term

For \(n=1\), substitute into the formula to find the first term: \( a_1 = (-1)^{1-1}(1+1) = 1 \times 2 = 2 \). So, the first term \(a_1 = 2\).
03

Find the Second Term

For \(n=2\), substitute into the formula to find the second term: \( a_2 = (-1)^{2-1}(2+1) = -1 \times 3 = -3 \). So, the second term \(a_2 = -3\).
04

Find the Third Term

For \(n=3\), substitute into the formula to find the third term: \( a_3 = (-1)^{3-1}(3+1) = 1 \times 4 = 4 \). So, the third term \(a_3 = 4\).
05

Find the Fourth Term

For \(n=4\), substitute into the formula to find the fourth term: \( a_4 = (-1)^{4-1}(4+1) = -1 \times 5 = -5 \). So, the fourth term \(a_4 = -5\).
06

Find the Fifth Term

For \(n=5\), substitute into the formula to find the fifth term: \( a_5 = (-1)^{5-1}(5+1) = 1 \times 6 = 6 \). So, the fifth term \(a_5 = 6\).

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Key Concepts

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

Alternating Sequences
An alternating sequence is a special type of mathematical sequence where the signs of the terms alternate between positive and negative. This can be seen in the formula \((-1)^{n-1}\), which is a key component in the provided sequence formula. The expression \((-1)^{n-1}\) determines whether each term is positive or negative.
  • When \(n\) is even, \((-1)^{n-1}\) becomes \(-1\), giving a negative term.
  • When \(n\) is odd, \((-1)^{n-1}\) becomes \(1\), giving a positive term.
Therefore, alternating sequences result in a repetitive pattern of signs which make them quite unique compared to sequences that purely decrease or increase. This pattern can have significant implications in mathematical scenarios, such as solving differential equations or analyzing signals in physics.
Linear Functions
Linear functions are functions that create a straight line when graphed. They are represented by the formula \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In the context of the sequence \(a_{n} = (-1)^{n-1}(n+1)\), the part \(n+1\) represents a linear function.
  • As \(n\) increases by 1, the term \(n+1\) increases linearly. This means that the value of each term changes at a constant rate.
  • The slope of this linear expression is 1, indicating that for every increase in \(n\), \(n+1\) also increases by 1.
Understanding the linear function part of this sequence helps in predicting how the numbers grow alongside the alternating sign pattern. It simplifies the process of calculating each sequence term.
Algebraic Expressions
An algebraic expression is a mathematical phrase that contains numbers, variables, and operations. In the sequence formula \(a_{n} = (-1)^{n-1}(n+1)\), both \((-1)^{n-1}\) and \(n+1\) are algebraic expressions.
  • The term \((-1)^{n-1}\) is a simple yet powerful algebraic expression that adjusts the sign of each sequence term, depending on whether \(n\) is odd or even.

  • The \(n+1\) part signifies a linear algebraic expression, contributing directly to the magnitude of each term, representing direct operations on the index \(n\).
By decomposing complex problems or sequences into algebraic expressions, you can analyze and solve each part systematically. It helps in understanding the sequential structure and ensures smooth calculations.

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Most popular questions from this chapter

Find the future value of each annuity. Payments of \(\$ 1000\) at the end of each year for 9 years at \(4 \%\) interest compounded annually

Suppose that \(a_{1}, a_{2}, a_{3}, \ldots\) and \(b_{1}, b_{2}, b_{3}, \ldots\) are arithmetic sequences. Let \(d_{n}=a_{n}+c \cdot b_{n},\) for any real number \(c\) and every positive integer \(n .\) Show that \(d_{1}, d_{2}, d_{3}, \ldots\) is an arithmetic sequence.

Height of a Dropped Ball Beth Schiffer drops a ball from a height of 10 meters and notices that on each bounce the ball returns to about \(\frac{3}{4}\) of its previous height. About how far will the ball travel before it comes to rest? (Hint: Consider the sum of two sequences.)

Depreciation in Value Each year a machine loses \(20 \%\) of the value it had at the beginning of the year. Find the value of the machine at the end of 6 years if it cost \(\$ 100,000\) new.

Find each sum that converges. $$\sum_{i=1}^{\infty}\left(\frac{1}{5}\right)\left(-\frac{1}{2}\right)^{i-1}$$
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