# How do you write a rule for the nth term of the arithmetic sequence and then find a_22 given -12, -5, 2, 9...?

Mar 17, 2018

color(green)(a_22 )= a_1 + (22-1) * d = -12 + (21 * 7) = color(green)(135

#### Explanation:

${a}_{1} = - 12$

${a}_{2} = - 5$

${a}_{2} - {a}_{1} = - 5 - \left(- 12\right) = - 5 + 12 = 7$

${a}_{3} - {a}_{2} = 2 - \left(- 5\right) = 2 + 5 = 7$

${a}_{4} - {a}_{3} = 9 - 2 = 7$

Note common difference $d = {a}_{4} - {a}_{3} = {a}_{3} - {a}_{2} = {a}_{2} - {a}_{1} = 7$

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

Similarly, ${a}_{4} = {a}_{3} + d = {a}_{1} + 2 d + d = {a}_{1} + \left(4 - 1\right) \cdot d$

In general, ${a}_{n} = {a}_{1} + \left(n - 1\right) \cdot d$

Hence ${a}_{22} = {a}_{1} + \left(22 - 1\right) \cdot d = - 12 + \left(21 \cdot 7\right) = 135$