# How do you determine whether the sequence ln1, ln2, ln3, ln4, ln5,... is arithmetic and if it is, what is the common difference?

Aug 22, 2017

the terms are not part of an AP

#### Explanation:

The term of an AP sequence takes the form:

$\left\{a , a + d , a + 2 d , a + 3 d , , \ldots\right\}$

Where:

$a =$first term
$d =$common difference

If the given sequence:

$\left\{\ln 1 , \ln 2 , \ln 3 , \ldots\right\}$

were an AP the clearly we have first term given by:

$a = \ln 1 \setminus \setminus \setminus \left(= 0\right)$

And the second term would satisfy:

$a + d = \ln 2$
$\implies \ln 1 + d = \ln 2$
$\implies d = \ln 2 - \ln 1 = \ln 2$

Thus, the third term would be given by:

$a + 2 d = \ln 1 + 2 \cdot \ln 2 = \ln 4 \ne \ln 3$

Hence, the terms are not part of an AP