Q3 Look for a pattern and predict t... [FREE SOLUTION]

91影视

Geometry

Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

$Math Studyset 91影视 Explanations$ Math

Student Edition 路 227 pages

ISBN: 9780395977279

Chapter 3: Q3 (page 107)

Look for a pattern and predict the next two numbers in each sequence.
$1, 鈥夆赌� \frac{1}{3}, 鈥夆赌� \frac{1}{9}, 鈥夆赌� \frac{1}{27}, 鈥夆赌� ....$

Short Answer

Expert verified

Sequence with next two numbers is $1, 鈥� \frac{1}{3}, 鈥� \frac{1}{9}, \frac{1}{27}, 鈥� \underset{炉}{\frac{1}{81}}, 鈥� \underset{炉}{\frac{1}{243}}, . ..$ .

Step by step solution

Step 1. Define deductive and inductive reasoning

Deductive reasoning is conclusions based on accepted statements like a theorem, postulate, facts etc.

Inductive reasoning is conclusions based on several past observations.

Step 2. Look for a pattern

While looking for a pattern in the sequence $1, 鈥� \frac{1}{3}, 鈥� \frac{1}{9}, 鈥� \frac{1}{27}, 鈥� ....$ reason inductively that is., we can conclude based on past observations.

Observe that,

$\begin{array}{l} 1, 鈥� \frac{1}{3}, 鈥� \frac{1}{9}, \frac{1}{27}, 鈥� ... \\ 鈮� \frac{1}{3^{0}}, 鈥� \frac{1}{3^{1}}, 鈥� \frac{1}{3^{2}}, 鈥� \frac{1}{3^{3}}, 鈥� ... \end{array}$

$\begin{array}{l} \frac{1}{3^{1}} 梅 \frac{1}{3^{0}} = \frac{1}{3} \\ \frac{1}{3^{2}} 梅 \frac{1}{3^{1}} = \frac{1}{3} \\ \frac{1}{3^{3}} 梅 \frac{1}{3^{2}} = \frac{1}{3} \end{array}$

So, the pattern of the sequence is that the next term is 1/3 times the current term.

Step 3. Predict the next two numbers of the sequence

So, the next two term of the sequence will be $\frac{1}{27} 脳 \frac{1}{3} = \frac{1}{81}$ and $\frac{1}{81} 脳 \frac{1}{3} = \frac{1}{243}$ .

Therefore, the sequence will be, $1, 鈥� \frac{1}{3}, 鈥� \frac{1}{9}, \frac{1}{27}, 鈥� \underset{炉}{\frac{1}{81}}, 鈥� \underset{炉}{\frac{1}{243}}, ...$ .