# Example:3 The 1st term of an arithmetic sequence is 200 and the common nunmber is -10 What is the formula an? What is the 20th term?

Jun 14, 2018

${a}_{n} = 200 + \left(n - 1\right) \cdot \left(- 10\right) = 200 - 10 \left(n - 1\right)$

${a}_{19} = 10$

#### Explanation:

An arithmetic sequence starts with an initial value and adds the same constant with every iteration. The terms are thus written like this:

${a}_{0} = {x}_{0}$
${a}_{1} = {x}_{0} + r$
${a}_{2} = {a}_{1} + r = \left({x}_{0} + r\right) + r = {x}_{0} + 2 r$
${a}_{3} = {a}_{2} + r = \left({x}_{0} + 2 r\right) + r = {x}_{0} + 3 r$
$\ldots$
${a}_{n} = {a}_{n - 1} + r = \left({x}_{0} + \left(n - 1\right) r\right) + r = {x}_{0} + n r$

In your case, the starting number ${x}_{0}$ is $200$, and the common number that we add every time is $- 10$

This means that the formula for the generic term is

${a}_{n} = 200 + \left(n - 1\right) \cdot \left(- 10\right) = 200 - 10 \left(n - 1\right)$

To find the $20$th term, just plug $n = 19$ in the generic equation. In fact, since we're starting from ${a}_{0}$, the indices are shifted so that ${a}_{0}$ is the first term, ${a}_{1}$ is the second, and so on. We get

${a}_{19} = 200 - 10 \cdot 19 = 200 - 190 = 10$