Question

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

$10,12,16,22,30,\mathrm{....}$

Short answer

Sequence with next two numbers is $10,12,16,22,30,\underset{¯}{40},\underset{¯}{52}\mathrm{...}$.

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 $10,12,16,22,30,\mathrm{....}$reason inductively that is., we can conclude based on past observations

As,

$\begin{array}{l}10+2=12\\ 12+4=16\\ 16+6=22\\ 22+8=30\end{array}$

So, the pattern of the sequence is that the even natural numbers 2, 4, 6, 8,.. and so on in order are added to each of the numbers to get the next number.

Step 3. Predict the next two numbers of the sequence.

Since, the pattern of the sequence is that the even natural numbers 2, 4, 6, 8,.. and so on in order are added to each of the numbers to get the next number.

So, the next two terms of the sequence will be,

$30+10=40$and $40+12=52$

Therefore, the sequence will be $10,12,16,22,30,\underset{¯}{40},\underset{¯}{52},\mathrm{....}$.