Look for a pattern and predict the next two numbers in each sequence. 1, 4, 16, 64, ....

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

Inductive reasoning is conclusions based on several past observations.

Here the given sequence can be written as,

$1,4,14,64,\mathrm{....}\approx 4^0,4^1,4^2,4^3,\mathrm{....}$

Therefore, while looking for a pattern we can reason inductively that is., we can conclude based on past observations. So, the pattern of the sequence is that the power of 4 is increasing by 1 in each term.

Since the power is increasing by 1, so, the next two term of the sequence according to the pattern is $4^4{\text{\hspace{0.33em}and4}}^5$which is equal to 256 and 1024.

So, the sequence will be, $1,4,16,64,\underset{¯}{256},\underset{¯}{1024},\mathrm{...}$.

Therefore, Sequence with next two numbers is $1,4,16,64,\underset{¯}{256},\underset{¯}{1024},...$.a