Answer
For an arithmetic sequence, $a_{n}=a_{n-1}+d.$
For a geometric sequence, $a_{n}=r\cdot a_{n-1}.$
Work Step by Step
For an arithmetic sequence, the terms $a_{n}$ and $a_{n-1}$ are such that
$ a_{n}-a_{n-1}=d,\qquad$
... They share a common difference between consecutive terms
... Adding $a_{n-1}$ to the equation, we obtain
$a_{n}=a_{n-1}+d$
For a geometric sequence, the terms $a_{n}$ and $a_{n-1}$ are such that
$\displaystyle \frac{a_{n}}{a_{n-1}}=r,\qquad$
... They share a common ratio between consecutive terms
... Multiplying the equation with $a_{n-1}$, we obtain
$a_{n}=r\cdot a_{n-1}$
