/*! 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 10 use inductive reasoning to find ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 10

use inductive reasoning to find the next two terms in each sequence. 1,2,4,8,16,32

Short Answer

Expert verified
The next two terms are 64 and 128.

Step by step solution

01

- Identify the Pattern

Observe the given sequence: 1, 2, 4, 8, 16, 32. Notice that each term is a power of 2. Specifically, the sequence can be written as: \( 2^0, 2^1, 2^2, 2^3, 2^4, 2^5 \).
02

- Generalize the Pattern

Recognize that the nth term of the sequence is given by the formula \( 2^{n-1} \).
03

- Find the Next Term

To find the next term in the sequence, substitute \( n = 6 \) (since the 6th term is \( 2^5 \)), which gives \( 2^6 = 64 \).
04

- Find the Following Term

To find the term after that, substitute \( n = 7 \), which gives \( 2^7 = 128 \).

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.

mathematical sequences
Mathematical sequences are lists of numbers arranged in a particular order following a specific rule. In the given sequence: 1, 2, 4, 8, 16, 32, the numbers increase in a pattern.
Understanding sequences is vital because they often appear in various fields of mathematics and science.
Sequences can be arithmetic (adding the same number each time), geometric (multiplying by the same number), or follow other rules.
In our example, the sequence is formed by multiplying the previous number by 2 every time, also known as a geometric sequence.
powers of 2
The concept of powers of 2 is crucial in understanding the given sequence.
A power of 2 means a number that can be written as 2 raised to the power of an integer. For instance,
  • The first term in the sequence is 1, which is equal to \(2^0\)
  • The second term is 2, which is \(2^1\)
  • The third term is 4, which is \(2^2\)
Each term in the sequence follows this pattern. Recognizing that these terms are all powers of 2 helps us understand and predict future terms easily.
So, the terms 32 and 64 can be written as \(2^5\)and \(2^6\).
pattern recognition
Pattern recognition is identifying recurring elements or trends within a dataset.
In sequences, pattern recognition involves finding a rule that connects each number with the next.
To determine the next terms in our sequence: 1, 2, 4, 8, 16, 32, we observe the multiplication pattern.
Every number is obtained by multiplying the previous one by 2. This repeated multiplication is the key pattern.
Using this pattern, the next term after 32 is \(32 \times 2=64\), and the term after that is \(64 \times 2 = 128\).
This illustrates that understanding patterns allows us to predict future values in the sequence.

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

Find the appropriate geometric model, and solve. If 40 houses in a community all need direct lines to one another in order to have telephone service, how many lines are necessary? Is that practical? Sketch and describe two models: first, model the situation in which direct lines connect every house to every other house and, second, model a more practical alternative.

Sketch, label, and mark the figure. A 3 -by-4-by-6-inch rectangular solid rests on its smallest face. Draw lines on the three visible faces to show how you can divide it into 72 identical smaller cubes.

Sketch and carefully label the figure. A net for a rectangular solid with the dimensions 1 by 2 by \(3 \mathrm{cm}\).

Think of a situation in which you have used inductive reasoning. Write a paragraph describing what happened and explaining why you think you used inductive reasoning.

Identify the statement as true or false. For each false statement, explain why it is false or sketch a counter example. A line segment that connects any two vertices in a polygon is called a diagonal.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Pure 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.