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

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$8,-4,2,-1,\frac{1}{2},\mathrm{....}$ reason inductively that is., we can conclude based on past observations.

Observe that,

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

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 $-\frac{1}{4}\times \left(-\frac{1}{2}\right)=\frac{1}{8}$.

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