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Given a sequence:

$$1, \frac12, \frac13, \frac14, \frac15,...$$

Its explicit formula can be given as:

$a(n) = \frac1n$ where $n \ge 1$.

I actually want to know is it a geometric sequence or an arithmetic one?

I tried finding common ratio and the common difference in this sequence to see if it's either one of them but was not successful.

My common ratio ($r$) and common difference ($d$) were some horrible values.

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    $\begingroup$ It's neither arithmetic nor geometric. $\endgroup$ Nov 1, 2016 at 3:02
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    $\begingroup$ It is a harmonic sequence, the reciprocals are in arithmetic progression. $\endgroup$ Nov 1, 2016 at 3:04
  • $\begingroup$ @AlexisOlson, Oh! I spent 4 hours on it to see :( Can you tell me what is the technique to see if a sequence lies either in geometric or an arithmetic one? Thanks! $\endgroup$
    – Anonymous
    Nov 1, 2016 at 3:04
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    $\begingroup$ @TitanicLover If the ratio of consecutive terms changes, then it is not geometric. If the difference of consecutive terms changes, then it is not arithmetic. $\endgroup$ Nov 1, 2016 at 3:10
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    $\begingroup$ To test $a_1,a_2,a_3,a_4......$ for arithmetic progression, take $a_2-a_1, a_3-a_2, a_4-a_3$, and check if all of them are the same. For GP take $a_2/a_1, a_3/a_2, a_4/a_3$ and check if all are the same. For HP(harmonic progressions) take $1/a_1, 1/a_2, 1/a_3, 1/a_4$ and check if all are in AP. $\endgroup$ Nov 1, 2016 at 3:10

2 Answers 2

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The sequence you gave is called the Harmonic sequence. It is neither geometric nor arithmetic. Not all sequences are geometric or arithmetic. For example, the Fibonacci sequence $1,1,2,3,5,8,...$ is neither.

A geometric sequence is one that has a common ratio between its elements. For example, the ratio between the first and the second term in the harmonic sequence is $\frac{\frac{1}{2}}{1}=\frac{1}{2}$. However, the ratio between the second and the third elements is $\frac{\frac{1}{3}}{\frac{1}{2}}=\frac{2}{3}$ so the common ratio is not the same and hence this is NOT a geometric sequence.

Similarly, an arithmetic sequence is one where its elements have a common difference. In the case of the harmonic sequence, the difference between its first and second elements is $\frac{1}{2}-1=-\frac{1}{2}$. However, the difference between the second and the third elements is $\frac{1}{3}-\frac{1}{2}=-\frac{1}{6}$ so the difference is again not the same and hence the harmonic sequence is NOT an arithmetic sequence.

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    $\begingroup$ "Not all sequences are geometric or arithmetic." — Understatement of the day ;-) $\endgroup$
    – celtschk
    Nov 1, 2016 at 6:56
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This is not a geometric series.

$a_1=1, a_2=\frac12, a_3=\frac13$

If this is a geometric sequence, then it is necessary that $\frac{a_2}{a_1}=\frac{a_3}{a_2}$

$$\frac{a_2}{a_1}=\frac{1}{2}$$

$$\frac{a_3}{a_2}=\frac{2}{3}$$

The two numbers are different, hence it is not a geometric sequence.

Similarly, you can verify that $$a_2-a_1 \neq a_3-a_2$$

To prove that something is a geometric sequence, you have to show that $\frac{a_{n+1}}{a_n}$ is a constant.
To prove that something is an arithmetic sequence, you have to show that $a_{n+1}-a_n$ is a constant.

For this problem, $$\frac{a_{n+1}}{a_n}=\frac{1/(n+1)}{1/n}=\frac{n}{n+1}=\frac{1}{1+1/n}$$ which is dependent on $n$.

while

$$a_{n+1}-a_n = \frac{1}{n+1}-\frac{1}{n}=-\frac{1}{n(n+1)}$$

which is again dependent on $n$.

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