The general term of a sequence is given a_{n} = (1/2)^{n}. 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.

arenceabigns 2021-03-08 Answered

The general term of a sequence is given \(a_{n} = \left(\frac{1}{2}\right)^{n}\). 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.

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tafzijdeq
Answered 2021-03-09 Author has 14057 answers

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

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