Question

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.

Polynomial arithmetic
ANSWERED
asked 2021-03-08

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.

Expert Answers (1)

2021-03-09

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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