Question

# 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,

Polynomial arithmetic

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.

2020-12-01
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.