foass77W

2020-11-30

The general term of a sequence is given ${a}_{n}={2}^{n}$. 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.

Aniqa O'Neill

If the subsequent terms in a series differ by a constant (common difference), the sequence is an arithmetic sequence (d).
nth term of an an arithmetic sequence ${a}_{n}=a+\left(n-1\right)d$

If there is a constant ratio between the terms in a series, the sequence is a geometric sequence.
nth term of an a geometric sequence ${a}_{n}=a\cdot {r}^{n}$

${a}_{n}={2}^{n}$
Substitute n = 0
${a}_{0}={2}^{0}$$=1$
Substitute n=1
${a}_{1}={2}^{1}$$=2$
Substitute n=2
${a}_{2}={2}^{2}$$=4$
Substitute n=3
${a}_{3}={2}^{3}$$=8$

The difference between first and second term is $2-1=1$.
The difference between second and third term is $4-2=2$. since, the successive terms is not differ by a constant. hence the sequence is not an arithmetic sequence.
The ratio of second term to first term is $\frac{2}{1}=2.$
The ratio of third term to second term is $\frac{4}{2}=2$.
The ratio of fourth term to third term is $\frac{8}{4}=2.$
The ratio between successive terms is constant. hence the sequence is a geometric sequence.

Do you have a similar question?