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Chapter 11: Problem 6

Write the first five terms of each sequence. $$a_{n}=\left(\frac{1}{2}\right)^{n}(n)$$

Short Answer

Expert verified
First five terms are \( \frac{1}{2}, \frac{1}{2}, \frac{3}{8}, \frac{1}{4}, \frac{5}{32} \).

Step by step solution

01

Understand the Formula

The given sequence is defined by the formula a_{n} = \left( \frac{1}{2} \right)^{n} (n)where \( n \) is the term number.
02

Find the First Term \( a_{1} \)

Substitute \( n = 1 \) into the sequence formula:\[\begin{align*}a_{1} &= \left( \frac{1}{2} \right)^{1} (1) \&= \frac{1}{2} \end{align*}\]
03

Find the Second Term \( a_{2} \)

Substitute \( n = 2 \) into the sequence formula:\[\begin{align*}a_{2} &= \left( \frac{1}{2} \right)^{2} (2) \&= \frac{1}{4} \times 2 \&= \frac{1}{2} \end{align*}\]
04

Find the Third Term \( a_{3} \)

Substitute \( n = 3 \) into the sequence formula:\[\begin{align*}a_{3} &= \left( \frac{1}{2} \right)^{3} (3) \&= \frac{1}{8} \times 3 \&= \frac{3}{8} \end{align*}\]
05

Find the Fourth Term \( a_{4} \)

Substitute \( n = 4 \) into the sequence formula:\[\begin{align*}a_{4} &= \left( \frac{1}{2} \right)^{4} (4) \&= \frac{1}{16} \times 4 \&= \frac{1}{4} \end{align*}\]
06

Find the Fifth Term \( a_{5} \)

Substitute \( n = 5 \) into the sequence formula:\[\begin{align*}a_{5} &= \left( \frac{1}{2} \right)^{5} (5) \&= \frac{1}{32} \times 5 \&= \frac{5}{32} \end{align*}\]

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Key Concepts

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

substituting values
Substituting values is a critical step in solving sequence problems. It involves placing specific numbers into a given formula to find the corresponding terms.
For the sequence formula given as: \[ a_n = \bigg(\frac{1}{2}\bigg)^n \big(n)\bigg) \]
you need to substitute the values of different terms into the formula.
Let's break it down to make it easy to understand:
  • Identify the term number: This is represented by 'n'. For instance, if you're finding the first term, 'n' equals 1.
  • Substitute 'n' into the formula: Replace 'n' with the term number in the formula. If you're finding the first term, the formula becomes: \[ a_{1} = \bigg(\frac{1}{2}\bigg)^{1} \times 1 \]
  • Calculate the answer: Simplify the equation to find the actual term value. For the first term, the calculation is: \[ a_{1} = \frac{1}{2}\bigg) \]

By understanding how to substitute values accurately, you can find different terms of any sequence with ease.
arithmetic sequences
An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. However, in our given formula: \[ a_{n} = \bigg(\frac{1}{2}\bigg)^{n} \times n \]
the sequence is not arithmetic because the difference between terms is not constant due to the exponent factor dependent on 'n'.
  • Constant Difference d: In arithmetic sequences, each term is the previous term plus a fixed value 鈥榙鈥�.
  • Formula for n-th term: This is expressed as: \[ a_n = a_1 + (n-1)d \] where \(a_1\) is the first term.
  • Example of arithmetic sequence: If your first term is 2 and the common difference is 3, the sequence would follow as: 2, 5, 8, 11, 14, etc.

Understanding arithmetic sequences helps in identifying patterns and simplifying problems that follow this format.
exponents
Exponents are used to express repeated multiplication of a number by itself. For the formula in the exercise: \[ a_n = \bigg(\frac{1}{2}\bigg)^{n} \times n \]
the exponent here is 'n', meaning 1/2 is raised to the power of 'n' for each term number:
  • Understanding Exponents: An exponent indicates how many times to multiply the base (in this case, 1/2) by itself. For example: \[ \bigg(\frac{1}{2}\bigg)^3 = \bigg(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\bigg) = \frac{1}{8} \]
  • Using Exponents in Sequences: Apply the power to the base and then multiply by 'n'. For the third term, the calculation is: \[ a_{3} = \bigg(\frac{1}{2}\bigg)^3 \times 3 = \frac{1}{8} \times 3 = \frac{3}{8} \]

The use of exponents allows simplification of repeated multiplications and is essential in understanding sequences involving exponential growth or decay.

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Most popular questions from this chapter

The table gives the results of a 2008 survey of Americans aged \(18-24\) in which the respondents were asked, "During the past 30 days, for about how many days have you felt that you did not get enough sleep?"$$\begin{array}{|l|c|c|c|c|}\hline \text { Number of Days } & 0 & 1-13 & 14-29 & 30 \\ \hline \text { Percent (as a decimal) } & 0.23 & 0.45 & 0.20 & 0.12 \\\\\hline\end{array}$$ Using the percents as probabilities, find the probability that, out of 10 respondents in the \(18-24\) age group selected at random, the following were true. Exactly 4 did not get enough sleep on \(1-13\) days.

Consider the word BRUCE. (a) In how many ways can all the letters of the word BRUCE be arranged? (b) In how many ways can the first 3 letters of the word BRUCE be arranged?

A financial analyst has determined the possibilities (and their probabilities) for the growth in value of a certain stock during the next year. (Assume these are the only possibilities.) See the table. For instance, the probability of a \(5 \%\) growth is \(0.15 .\) If you invest \(\$ 10,000\) in the stock, what is the probability that the stock will be worth at least \(\$ 11,400\) by the end of the year? $$\begin{array}{c|c}\hline \text { Percent Growth } & \text { Probability } \\\\\hline 5 & 0.15 \\\\\hline 8 & 0.20 \\\\\hline 10 & 0.35 \\\\\hline 14 & 0.20 \\\\\hline 18 & 0.10 \\\\\hline\end{array}$$

Assume that \(n\) is a positive integer. Use mathematical induction to prove each statement S by following these steps. See Example \(I\). (a) Verify the statement for \(n=1\) (b) Write the statement for \(n=k\) (c) Write the statement for \(n=k+1\) (d) Assume the statement is true for \(n=k\). Use algebra to change the statement in part (b) to the statement in part (c). (e) Write a conclusion based on Steps (a)-(d). $$5+10+15+\dots+5 n=\frac{5 n(n+1)}{2}$$

Use the summation properties and rules to evaluate each series. $$\sum_{i=1}^{5}\left(4 i^{2}-2 i+6\right)$$
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