Step 1

A sequence is is an Arithmetic sequence if the successive terms is differ by constant, called common difference (d).

nth term of an an arithmetic sequence is given by formula,

\(a_{n} = a + ( n − 1)d\)

A sequence is is a Geometric sequence if the ratio between successive terms is constant.

nth term of an a geometric sequence is given by formula,

\(a_{n} = a \cdot r^{n}\)

Step 2

Consider the given sequence,

\(a_{n} = 2^{n}

Substitute n = 0 to find the first term of this sequence,

\(a_{0} = 2^{0}\)

\(=1\)

Substitute n=1 to find the second term of this sequence,

\(a_{1} = 2^{1}\)

\(=2\)

Substitute n=2 to find the third term of this sequence,

\(a_{2} = 2^{2}\)

\(=4\)

Substitute n=3 to find the fourth term of this sequence,

\(a_{3} = 2^{3}\)

\(=8\)

so the sequence is 1, 2, 4, 8

Step 3

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.

Now, 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 \(\frac{8}{4}=2.\)

Since, the ratio between successive terms is constant. hence the sequence is a geometric sequence.

A sequence is is an Arithmetic sequence if the successive terms is differ by constant, called common difference (d).

nth term of an an arithmetic sequence is given by formula,

\(a_{n} = a + ( n − 1)d\)

A sequence is is a Geometric sequence if the ratio between successive terms is constant.

nth term of an a geometric sequence is given by formula,

\(a_{n} = a \cdot r^{n}\)

Step 2

Consider the given sequence,

\(a_{n} = 2^{n}

Substitute n = 0 to find the first term of this sequence,

\(a_{0} = 2^{0}\)

\(=1\)

Substitute n=1 to find the second term of this sequence,

\(a_{1} = 2^{1}\)

\(=2\)

Substitute n=2 to find the third term of this sequence,

\(a_{2} = 2^{2}\)

\(=4\)

Substitute n=3 to find the fourth term of this sequence,

\(a_{3} = 2^{3}\)

\(=8\)

so the sequence is 1, 2, 4, 8

Step 3

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.

Now, 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 \(\frac{8}{4}=2.\)

Since, the ratio between successive terms is constant. hence the sequence is a geometric sequence.