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
ANSWERED
asked 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.

Expert Answers (1)

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.
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