Question

Indicate the general rule for the arithmetic sequence with a3 = -12 and a8 = -37. an = -2 + (n-1)(-5) an = -2 + (n-1)(5) an = 2 + (n-1)(-5) an = 2 + (n-1)(5)

Answer 1

From the 3rd term to the 8th term, we'd need to use the "common difference" or "d", 5 times, namely add it that many times, to get the 8th term, thus

$\bf \begin{array}{ll} term&value\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ a_3&-12\\ a_4&-12+d\\ a_5&-12+d+d\\ a_6&-12+d+d+d\\ a_7&-12+d+d+d+d\\ a_8&-12+d+d+d+d+d\\ &-12+5d\\ &-37 \end{array}\implies -12+5d=-37 \\\\\\ 5d=-25\implies d=\cfrac{-25}{5}\implies d=-5$

we know the 3rd term is -12, so the first one has to be -12+5+5, undoing the -5's, so the first term is -2.
 
$\bf n^{th}\textit{ term of an arithmetic sequence} \\\\ a_n=a_1+(n-1)d\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ d=\textit{common difference}\\ ----------\\ a_1=-2\\ d=-5 \end{cases} \\\\\\ a_n=-2+(n-1)(-5)$