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Chapter 14: Problem 10

Write out the first five terms of each sequence. $$ a_{n}=6(-1)^{n+1} $$

Short Answer

Expert verified
The first five terms are 6, -6, 6, -6, 6.

Step by step solution

01

- Understand the formula

The general term of the sequence is given by \( a_n = 6(-1)^{n+1} \). This means the value of each term depends on the position \( n \) in the sequence.
02

- Calculate the first term

For \( n = 1 \), \( a_1 = 6(-1)^{1+1} = 6(-1)^2 = 6 \cdot 1 = 6 \). So the first term \( a_1 = 6 \).
03

- Calculate the second term

For \( n = 2 \), \( a_2 = 6(-1)^{2+1} = 6(-1)^3 = 6 \cdot (-1) = -6 \). So the second term \( a_2 = -6 \).
04

- Calculate the third term

For \( n = 3 \), \( a_3 = 6(-1)^{3+1} = 6(-1)^4 = 6 \cdot 1 = 6 \). So the third term \( a_3 = 6 \).
05

- Calculate the fourth term

For \( n = 4 \), \( a_4 = 6(-1)^{4+1} = 6(-1)^5 = 6 \cdot (-1) = -6 \). So the fourth term \( a_4 = -6 \).
06

- Calculate the fifth term

For \( n = 5 \), \( a_5 = 6(-1)^{5+1} = 6(-1)^6 = 6 \cdot 1 = 6 \). 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.

general term
In any sequence of numbers, the general term is critical for understanding the pattern that creates the sequence. The general term is represented by a formula, denoted as \(a_n\). For example, in our sequence, the general term is \(a_n = 6(-1)^{n+1}\). This formula tells us how to calculate the value of the sequence at any position \(n\). To use it, we simply substitute \(n\) with the desired position number. Knowing the general term helps us understand the overall structure of the sequence without having to write out every single term.
alternating sequence
An alternating sequence is a sequence where the terms switch back and forth between positive and negative values. In the example given, the sequence alternates between 6 and -6. This pattern occurs because of the term \((-1)^{n+1}\) in the general term formula. When \(n\) is an odd number, \( (-1)^{n+1} \) becomes positive because the power of \((-1)\) is even. When \(n\) is an even number, \( (-1)^{n+1} \) becomes negative because the power of \((-1)\) is odd. This regular switching between positives and negatives characterizes an alternating sequence.
powers of negative one
Understanding powers of negative one is essential when dealing with alternating sequences. The term \((-1)^{n+1}\) means that we raise -1 to the power of \(n+1\). Key points to remember:
  • If the exponent (the power) is even, like 2, 4, or 6, the result will always be +1.
  • If the exponent is odd, like 1, 3, or 5, the result will always be -1.

In our example, since the exponent \(n+1\) changes with each \(n\), the result alternates between +1 and -1. This alternation is what makes the sequence alternate between 6 and -6. Working with powers of negative one helps us predict whether a term in the sequence will be positive or negative.

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Most popular questions from this chapter

Solve each applied problem by writing the first few terms of a sequence. Leslie Maruri is offered a new modeling job with a salary of \(20,000+2500 n\) dollars per year at the end of the \(n\) th year. Write a sequence showing her salary at the end of each of the first 5 yr. If she continues in this way, what will her salary be at the end of the tenth year?

Solve each application. (Hint: Immediately after reading the problem, determine whether you need to find a specific term of a sequence or the sum of the terms of a sequence.) When dropped from a certain height, a ball rebounds \(\frac{3}{5}\) of the original height. How high will the ball rebound after the fourth bounce if it was dropped from a height of \(10 \mathrm{ft} ?\) (IMAGE CANNOT COPY)

Find the indicated term of each binomial expansion. See Example 6 The term with \(x^{8} y^{2}\) in \(\left(2 x^{2}+3 y\right)^{6}\)

Solve each applied problem by writing the first few terms of a sequence. A certain car loses \(\frac{1}{2}\) of its value each year. If this car cost \(\$ 40,000\) new, what is its value at the end of 6 yr?

Evaluate \(a+(n-1) d\) for \(a=-2, n=5,\) and \(d=3\)
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