/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 12 Write recursive routines to help... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 12

Write recursive routines to help you answer \(12 \mathrm{a}-\mathrm{d}\). a. Find the 9 th term of \(1,3,9,27, \ldots\) (a) b. Find the 123rd term of \(5,-5,5,-5, \ldots\) (it) c. Find the term number of the first positive term of the sequence \(-16.2,-14.8\), \(-13.4,-12, \ldots .\) d. Which term is the first to be either greater than 100 or less than \(-100\) in the sequence \(-1,2,-4,8,-16, \ldots\) ?

Short Answer

Expert verified
a) 6561, b) 5, c) 13, d) 8.

Step by step solution

01

Identify if it's a Geometric Sequence (a)

In a geometric sequence, each term after the first is found by multiplying the previous one by a constant. Here, each term is multiplied by 3.The formula for the nth term of a geometric sequence is given by: \( a_n = a_1 imes r^{(n-1)} \), where \( a_1 \) is the first term and \( r \) is the common ratio.
02

Apply the Formula for Part a

Given the sequence \(1, 3, 9, 27, \ldots\), we identify \( a_1 = 1 \) and \( r = 3 \). To find the 9th term:\[ a_9 = 1 imes 3^{(9-1)} = 3^8 = 6561 \]
03

Determine Sequence Pattern (b)

The sequence \(5, -5, 5, -5, \ldots\) is an alternating sequence of two terms, meaning every term is either 5 or -5.Even terms (2nd, 4th, 6th, etc.) are \(-5\) and odd terms (1st, 3rd, 5th, etc.) are \(5\).
04

Find the 123rd Term of Alternating Sequence

Since 123 is an odd number, the 123rd term will be the same as the first term, which is \(5\).
05

Find the First Positive Term (c)

The sequence is arithmetic, decreasing by 1.4 each time. The general term is given by:\( a_n = a_1 + (n-1) imes d \), where \( a_1 = -16.2 \) and \( d = 1.4 \).Setting \( a_n > 0 \) to find the first positive term:\[ -16.2 + (n-1) imes 1.4 > 0 \].
06

Solve the Inequality for Part c

Solve the inequality:\[ (n-1) imes 1.4 > 16.2 \]\[ n - 1 > \frac{16.2}{1.4} \]\[ n - 1 > 11.57 \]\[ n > 12.57 \].The smallest integer \( n \) is 13.
07

Find First Term Greater Than 100 or Less Than -100 (d)

The sequence \(-1, 2, -4, 8, -16, \ldots\) is a geometric sequence with initial term \( a_1 = -1 \) and ratio \( r = -2 \).The nth term is \( a_n = -1 \times (-2)^{(n-1)} \). Find \( n \) where \( |a_n| > 100 \).
08

Solve Inequality for Part d

We need \(|a_n| > 100\), meaning:\[ |-1 imes (-2)^{(n-1)}| > 100 \]\[ 2^{(n-1)} > 100 \].Solving for \( n \):\[ n-1 > \log_2{100} \approx 6.644 \]\[ n > 7.644 \].The smallest integer \( n \) is 8.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Geometric Sequence
A geometric sequence is a number series created by multiplying the previous term by a fixed, non-zero number termed as the common ratio. The exponential growth or decay pattern makes geometric sequences unique. For instance, in the sequence 1, 3, 9, 27, each number is produced by multiplying the prior number by 3. The formula to determine the nth term is:
  • \( a_n = a_1 \times r^{(n-1)} \)
where:
  • \( a_1 \) = first term
  • \( r \) = common ratio
  • \( n \) = term number
Using this, you find the 9th term of a sequence like 1, 3, 9, 27 by substituting these values into the formula.
Arithmetic Sequence
An arithmetic sequence, unlike geometric, adds or subtracts a consistent value, known as the common difference. It's a simple linear pattern, seen in sequences like -16.2, -14.8, -13.4, where each term decreases by 1.4. The formula for the nth term is:
  • \( a_n = a_1 + (n-1) \times d \)
where:
  • \( a_1 \) = first term
  • \( d \) = common difference
  • \( n \) = term number
You can use this formula to find any specific term by setting the terms into this equation, ideal for detecting positive term occurrences.
Alternating Sequence
Alternating sequences switch a pattern back and forth, adding a twist to series progression. In the sequence 5, -5, 5, -5, each term alternates between positive and negative. Recognize the pattern or formula for specific terms:
  • Odd-numbered terms are the same as the first term
  • Even-numbered terms are negative of the odd terms
By understanding the sequence makeup, determining any specific term, like the 123rd in this case, becomes straightforward. Since 123 is odd, it repeats the first term.
Inequality Solving
Inequality solving is crucial to understand sequences better, particularly for determining when terms exceed certain bounds. Solving inequalities involves finding conditions under which certain qualities hold, such as in the sequence -1, 2, -4, 8, where you might seek when a term surpasses 100 in absolute value.
  • Use the general term formula, such as \( a_n = a_1 \times r^{(n-1)} \)
  • Apply inequality logic: \( |a_n| > k \) to find the required terms
Like in finding which term exceeds an absolute value of 100, recognizing when the term \( 2^{(n-1)} > 100 \) helps find the smallest suitable \( n \). Solving these inequalities can pinpoint markers like when values become significantly large.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Run the easy level of the LINES program on your calculator. [ \(\ \square\) See Calculator Note 3D to learn how to use LINES program. 4] Sketch a graph of the randomly generated line on your paper. Use the trace function to locate the \(y\)-intercept and to determine the rate of change. When the calculator says you have the correct equation, write it under the graph. Repeat this program until you get three correct equations in a row.

Match these walking instructions with their graph sketches. a. The walker stands still. b. The walker takes a few steps toward the 0 -mark, then walks away. c. The walker steps away from the 0 -mark, stops, then continues more slowly in the same direction.

Undo the order of operations to find the \(x\)-value in each equation. a. \(3(x-5.2)+7.8=14\) (a) b. \(3.5\left(\frac{x-8}{4}\right)=2.8\)

On his Man in Motion World Tour in 1987, Canadian Rick Hansen wheeled himself \(24,901.55\) miles to support spinal cord injury research and rehabilitation, and wheelchair sport. He covered 4 continents and 34 countries in two years, two months, and two days. Learn more about Rick's journey with the link. at mww.kevmath.comiDA. a. Find Rick's average rate of travel in miles per day. (Assume there are 365 days in a year and \(30.4\) days in a month.) (Tit) b. How much farther would Rick have traveled if he had continued his journey for another \(1 \frac{1}{2}\) years? c. If Rick continued at this same rate, how many days would it take him to travel 60,000 miles? How many years is that?

Jo mows lawns after school. She finds that she can use the equation \(P=-300+15 \mathrm{~N}\) to calculate her profit. a. Give some possible real-world meanings for the numbers \(-300\) and 15 and the variable \(N\). b. Invent two questions related to this situation and then answer them. c. Solve the equation \(P=-300+15 N\) for the variable \(N\). d. What does the equation in \(7 \mathrm{c}\) tell you?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.