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Chapter 8: Problem 5

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=\left(\frac{1}{3}\right)^{n}(n-1)$$

Short Answer

Expert verified
The first five terms are: 0, \(\frac{1}{9}\), \(\frac{2}{27}\), \(\frac{1}{27}\), \(\frac{4}{243}\).

Step by step solution

01

Understanding the Sequence Formula

The sequence formula is given as \( a_n = \left(\frac{1}{3}\right)^n (n-1) \). This formula directs us to substitute different values for \( n \) to compute each term of the sequence.
02

Finding the First Term \( a_1 \)

Substitute \( n = 1 \) into the sequence formula: \[ a_1 = \left(\frac{1}{3}\right)^1 (1-1) = \left(\frac{1}{3}\right)^1 \times 0 = 0 \]. The first term, \( a_1 \), is 0.
03

Finding the Second Term \( a_2 \)

Substitute \( n = 2 \) into the sequence formula: \[ a_2 = \left(\frac{1}{3}\right)^2 (2-1) = \left(\frac{1}{3}\right)^2 \times 1 = \frac{1}{9} \]. The second term, \( a_2 \), is \( \frac{1}{9} \).
04

Finding the Third Term \( a_3 \)

Substitute \( n = 3 \) into the sequence formula: \[ a_3 = \left(\frac{1}{3}\right)^3 (3-1) = \left(\frac{1}{3}\right)^3 \times 2 = \frac{2}{27} \]. The third term, \( a_3 \), is \( \frac{2}{27} \).
05

Finding the Fourth Term \( a_4 \)

Substitute \( n = 4 \) into the sequence formula: \[ a_4 = \left(\frac{1}{3}\right)^4 (4-1) = \left(\frac{1}{3}\right)^4 \times 3 = \frac{3}{81} = \frac{1}{27} \]. The fourth term, \( a_4 \), is \( \frac{1}{27} \).
06

Finding the Fifth Term \( a_5 \)

Substitute \( n = 5 \) into the sequence formula: \[ a_5 = \left(\frac{1}{3}\right)^5 (5-1) = \left(\frac{1}{3}\right)^5 \times 4 = \frac{4}{243} \]. The fifth term, \( a_5 \), is \( \frac{4}{243} \).

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Key Concepts

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

Terms of a Sequence
A sequence is a set of numbers arranged in a specific order, where each number is called a term. When given a sequence formula, like \( a_n = \left(\frac{1}{3}\right)^n (n-1) \), it helps us find the terms of the sequence by substituting integer values starting from 1, 2, 3, and so on. Each value of \( n \) gives us a different term:
  • The first term, \( a_1 \), results from substituting \( n = 1 \).
  • The second term, \( a_2 \), is found by setting \( n = 2 \).
  • Continue in this pattern for subsequent terms.
This systematic approach ensures that every term can be found without ambiguity. Each term builds on the previous one to form a sequence that might show a particular behavior or pattern, such as increasing, decreasing, or oscillating.
Exponential Function
An exponential function is a type of mathematical function expressed in the form \( a^n \), where \( a \) is a constant and \( n \) is a variable exponent. In the context of our sequence, the exponential part \( \left(\frac{1}{3}\right)^n \) plays a significant role in defining each term.
  • As \( n \) increases, this function causes each term to become smaller because the base, \( \frac{1}{3} \), is between 0 and 1.
  • Exponential decay is showcased as successive terms have increasingly smaller values.
This attribute of the exponential function creates an effect where each subsequent term is a fraction of the previous one, demonstrating the power of exponential functions in modeling such behaviors.
Algebra
Algebra is a field of mathematics that helps us manipulate symbols and formulas to solve problems. When dealing with sequences, algebraic operations allow us to substitute values and simplify expressions.
  • For our sequence formula \( a_n = \left(\frac{1}{3}\right)^n (n-1) \), algebra involves substituting a chosen \( n \), performing the exponentiation, and multiplying by \((n-1)\).
  • Simplifying the result often requires further algebraic manipulation, like reducing fractions.
  • Algebraic reasoning helps validate that each computed term is correct.
By understanding algebraic principles, we can easily navigate through sequences and perform necessary calculations accurately, reinforcing our comprehension in mathematics.

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Most popular questions from this chapter

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{4}=5, d=-2$$

Use a graphing calculator to evaluate each sum. Round to the nearest thousandth. $$\sum_{i=1}^{10}-(1.4)^{i}$$

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{1}=6, a_{2}=3$$

Swing of a Pendulum A pendulum bob swings through an arc 40 centimeters long on its first swing. Each swing thereafter, it swings only \(80 \%\) as far as on the previous swing. How far will it swing altogether before coming to a complete stop?

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{1}=-3, d=-4$$
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