Question

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

$8,-4,2,-1,\frac{1}{2},\mathrm{....}$

Short answer

Sequence with next two numbers is $\mathbf{8}\mathbf{,}\mathbf{-}\mathbf{4}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{-}\mathbf{1}\mathbf{,}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\underset{\mathbf{¯}}{\mathbf{-}\frac{\mathbf{1}}{\mathbf{4}}}\mathbf{,}\underset{\mathbf{¯}}{\frac{\mathbf{1}}{\mathbf{8}}}\mathbf{,}\mathbf{.}\mathbf{..}$.

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

While looking for a pattern in the sequence $8,-4,2,-1,\frac{1}{2},\mathrm{....}$reason inductively that is., we can conclude based on past observations.

$\begin{array}{c}8\times \left(-\frac{1}{2}\right)=-4\\ -4\times \left(-\frac{1}{2}\right)=2\\ 2\times \left(-\frac{1}{2}\right)=-1\\ -1\times \left(-\frac{1}{2}\right)=\frac{1}{2}\end{array}$

Observe that,

So, the pattern of the sequence is that the next term is $\left(-\frac{1}{2}\right)$times the previous term.

Step 3. Predict the next two numbers of the sequence

So, the next two term of the sequence will be $\frac{1}{2}\times \left(-\frac{1}{2}\right)=-\frac{1}{4}$and $-\frac{1}{4}\times \left(-\frac{1}{2}\right)=\frac{1}{8}$.