# How do you write a rule for the nth term of the arithmetic sequence a_7=21, a_13=42?

Apr 11, 2017

${a}_{n} = \frac{7}{2} \left(n - 1\right)$

#### Explanation:

Arithmetic sequence, ${a}_{n} = {a}_{1} + \left(n - 1\right) d$, d=common difference.

given that,
${a}_{7} = 21$
${a}_{1} + \left(7 - 1\right) d = 21$
${a}_{1} + 6 d = 21 \to$A

${a}_{13} = 42$
${a}_{1} + 12 d = 42 \to$B

B-A,
$6 d = 21$ $\to d = \frac{21}{6} = \frac{7}{2}$

Plug in $d = \frac{7}{2}$ in A or B.
${a}_{1} + 6 \left(\frac{7}{2}\right) = 21$
${a}_{1} + 21 = 21$ $\to {a}_{1} = 0$

therefore,
${a}_{n} = 0 + \left(n - 1\right) \frac{7}{2}$
${a}_{n} = \frac{7}{2} \left(n - 1\right)$