Step 1

To Determine: whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference, if it is geometric, find the common ratio.

Given: we have the general term of a sequence

\(a_{n}=\left(\frac{1}{2}\right)^{n}\)

Explanation: we have

\(a_{n}=\left(\frac{1}{2}\right)^{n}\)

now we can write down the sequence by putting \(n=1,2,3....\)

so we have \(\frac{1}{2}, \frac{1}{2}^{2}, \frac{1}{2}^{3}, \frac{1}{2}^{4} ...\)

this sequence is G.P because this give same common ratio \(\frac{1}{2}\) as follows

Step 2

\(r1=\frac{1}{2}^{\frac{21}{2}}=\frac{1}{2}\)

\(r2=\frac{1}{2}^{\frac{31}{2}^{2}}=\frac{1}{2}\)

\(r3=\frac{1}{2}^{\frac{41}{2}^{3}}=\frac{1}{2}\)

and so on.hence the common ratio will be \(\frac{1}{2}\)

Answer:

\(\text{first term}\ a=\frac{1}{2} \text{and common ratio}\ r=\frac{1}{2}\).