Use inductive reasoning to predict the next number in each sequence.

1,4,2,8,4,16,8,32,...

Inductive reasoning is conclusions based on several past observations.

While looking for a pattern in the sequence 1,4,2,8,4,16,8,32,... reason inductively that is., we can conclude based on past observations

As,

$$\begin{gathered} a_1=1 \\ {a}_2=4 \\ {a}_3=2a_1=2 \\ {a}_4=2a_2=8 \\ {a}_5=2a_3=4 \\ {a}_6=2a_4=16 \\ {a}_7=2a_5=8 \\ {a}_8=2a_6=32 \end{gathered}$$

So, the pattern of the sequence is that two alternate sequences are there {1,2,4,8…} and {4,8,16,32}, wherein each term is getting multiplied by 2 to obtain next term.

Since, the pattern of the sequence is that two alternate sequences are there {1,2,4,8…} and {4,8,16,32}, wherein each term is getting multiplied by 2 to obtain next term., therefore, the next term of the sequence will be,

$a_9=2a_7=16$.

Therefore, the sequence will be $1,4,2,8,4,16,8,32,\underset{¯}{16}\mathrm{...}$.