Chapter 3: Q3 (page 107)

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

Short Answer

Expert verified

Sequence with next two numbers is $1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \underline{\frac{1}{81}}, \underline{\frac{1}{243}}, . ..$ .

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 $1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, ....$ reason inductively that is., we can conclude based on past observations.

Observe that,

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

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

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 $\frac{1}{27} \times \frac{1}{3} = \frac{1}{81}$ and $\frac{1}{81} \times \frac{1}{3} = \frac{1}{243}$ .

Therefore, the sequence will be, $1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \underline{\frac{1}{81}}, \underline{\frac{1}{243}}, ...$ .

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

Short Answer

Step by step solution

Step 1. Define deductive and inductive reasoning

Step 2. Look for a pattern

Step 3. Predict the next two numbers of the sequence

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Look for a pattern and predict the next two numbers in each sequence.1, 13, 19, 127, ....

Short Answer

Step by step solution

Step 1. Define deductive and inductive reasoning

Step 2. Look for a pattern

Step 3. Predict the next two numbers of the sequence

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

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