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.

Question
Polynomial arithmetic
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.

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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