foass77W

2020-11-30

The general term of a sequence is given

If the sequence is arithmetic, find the common difference, if it is geometric, find the common ratio.

Aniqa O'Neill

Skilled2020-12-01Added 100 answers

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 ${a}_{n}=a+(n-1)d$

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.

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