Question

Indicate the general rule for the arithmetic sequence with a3 = -4 and a8 = -29.

Answer 1

The general rule would be

$a_n=6-5(n-1)$

We first find the common difference, the value that is added every time to find the next term.  This is the same as the slope; the formula for slope is

m=(y₂-y₁)/(x₂-x₁)

Letting x be the term number and y be the term value, we have
m=(-29--4)/(8-3) = (-29+4)/(8-3) = -25/5 = -5

Now we work backwards from a₃ to find a₁, the first term.  We know that the common difference is -5; that means each time, the next term is found by subtracting 5.  If we want to work backward we will add 5:
-4+5=1 for a₂
1+5=6 for a₁

Now we can write the general form:
$a_n=6-5(n-1)$