Question

Look for a pattern and predict the next two numbers in each sequence.

$1,\frac{1}{3},\frac{1}{9},\frac{1}{27},\mathrm{....}$

Short answer

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

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

Observe that,

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

As,

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

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, $1,\frac{1}{3},\frac{1}{9},\frac{1}{27},\underset{¯}{\frac{1}{81}},\underset{¯}{\frac{1}{243}},\mathrm{...}$.