# How do you write a rule for the nth term of the arithmetic sequence given a_8=-44, a_5=-32?

Mar 17, 2018

Therefore color(brown)(a_n = -12 - 4n = -4(3 + n)

#### Explanation:

${a}_{8} - {a}_{5} = - 44 - + 32 = - 12$

But ${a}_{8} - {a}_{5} = 3 \cdot d$ where d is the common difference between successive terms.

$\therefore 3 d = - 12$ or $d = - 4$

${a}_{2} = {a}_{1} + d$

${a}_{3} = {a}_{2} + d = {a}_{1} + 2 d = {a}_{1} + \left(3 - 1\right)$

Likewise, ${a}_{5} = {a}_{4} + d = {a}_{1} + \left(5 - 1\right) \cdot d = {a}_{1} + \left(4 \cdot - 4\right) = {a}_{1} - 16$

Hence, ${a}_{1} = {a}_{5} + 16 = - 32 + 16 = - 16$

${a}_{1} = - 16 , d = - 4$

Therefore ${a}_{n} = {a}_{1} + \left(n - 1\right) \cdot d = - 16 + \left(n - 1\right) \cdot \left(- 4\right)$

$\implies - 16 - 4 n + 4 = - 12 - 4 n$