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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 \( 2^7 = 128 \).

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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.

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Most popular questions from this chapter

Write the first five terms of two different sequences in which 12 is the 3rd term.

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

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