Question

Look for a pattern and predict the next two numbers in each sequence. $1,4,9,16,\mathrm{....}$

Short answer

Sequence with next two numbers is $\mathbf{1}\mathbf{,}\mathbf{4}\mathbf{,}\mathbf{9}\mathbf{,}\mathbf{16}\mathbf{,}\underset{\mathbf{¯}}{\mathbf{25}}\mathbf{,}\underset{\mathbf{¯}}{\mathbf{36}}\mathbf{,}\mathbf{.}\mathbf{..}$.

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

Here the given sequence can be written as,

$1,4,9,16,\mathrm{....}\approx 1^2,2^2,3^2,4^2,\mathrm{....}$

Therefore, while looking for a pattern we can reason inductively that is., we can conclude based on past observations. So, the following numbers are the sequence of numbers.

Step 3. Predict the next two numbers of the sequence

Since the sequence is squares of natural numbers, so, the next two term of the sequence according to the pattern is $5^2{\text{\hspace{0.33em}and6}}^2$which is equal to 25 and 36.

So, the sequence will be, $1,4,16,64,\underset{¯}{25},\underset{¯}{36},\mathrm{...}$.

Therefore, sequence with next two numbers is $1,4,16,64,\underset{¯}{25},\underset{¯}{36},\mathrm{...}$..