# Is the the sequence 5a-1, 3a-1, a-2, -a-1,... arithmetic?

Dec 11, 2016

Interpretation 1: Assuming the question is with $a - 2$:

The common difference, $d$, of a sequence will determine whether it is arithmetic, geometric or neither. If it is arithmetic, the common difference will be a common number added or subtracted from the previous term.

$d = {t}_{2} - {t}_{1}$

$d = 3 a - 1 - \left(5 a - 1\right)$

$d = 3 a - 1 - 5 a + 1$

$d = - 2 a$

$d = {t}_{3} - {t}_{2}$

$d = a - 2 - \left(3 a - 1\right)$

$d = - 2 a - 1$

Since the two numbers aren't the same, this sequence is not arithmetic

Interpretation 2: Assuming the question is with $a - 1$

Doing the same process as above:

$d = {t}_{2} - {t}_{1}$

$d = 3 a - 1 - \left(5 a - 1\right)$

$d = 3 a - 1 - 5 a + 1$

$d = - 2 a$

$d = {t}_{3} - {t}_{2}$

$d = a - 1 - \left(3 a - 1\right)$

$d = a - 1 - 3 a + 1$

$d = - 2 a$

Since the two $d ' s$ are the same, this sequence is arithmetic.

Hopefully this helps!