# How do I find the nth term of an arithmetic sequence?

Mar 25, 2016

${a}_{n} = {a}_{1} + \left(n - 1\right) \cdot d$
$\textcolor{w h i t e}{\text{XXX}}$where ${a}_{1}$ is the first term and
$\textcolor{w h i t e}{\text{XXXXXXX}} d$ is the difference between a term and its previous term.

#### Explanation:

Examine the pattern:

${a}_{1}$
${a}_{\textcolor{b r o w n}{2}} = {a}_{1} + d = \textcolor{g r e e n}{{a}_{1} + 1 d}$
${a}_{\textcolor{b r o w n}{3}} = {a}_{2} + d = {a}_{1} + d + d = \textcolor{g r e e n}{{a}_{1} + 2 d}$
${a}_{\textcolor{b r o w n}{4}} = {a}_{3} + d = {a}_{1} + 2 d + \mathrm{dc} o l \mathmr{and} \left(g r e e n\right) \left(= {a}_{1} + 3 d\right)$
${a}_{\textcolor{b r o w n}{5}} = {a}_{4} + d = {a}_{1} + 3 d + d = \textcolor{g r e e n}{{a}_{1} + 4 d}$