arenceabigns

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.

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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 $\frac{1}{2}$