# How do you find the formula for a_n for the arithmetic sequence a_5=190, a_10=115?

Feb 13, 2017

See explanation.

#### Explanation:

From the given data we can easily calculate the difference $d$ of the sequence.

If ${a}_{5} = {a}_{1} + 4 \cdot d$ and ${a}_{10} = {a}_{1} + 9 \cdot d$, the we can calculate that:

${a}_{10} - {a}_{5} = 5 \cdot d$, so: $d = \frac{{a}_{10} - {a}_{5}}{5}$

$d = \frac{115 - 190}{5} = - \frac{75}{5} = - 15$

Now we can calculate ${a}_{1}$ using:

${a}_{5} = {a}_{1} + 4 \cdot d$

$190 = {a}_{1} + 4 \cdot \left(- 15\right)$

$190 = {a}_{1} - 60$

${a}_{1} = 250$

So finally the $n - t h$ term is:

${a}_{n} = 250 + \left(n - 1\right) \cdot \left(- 15\right)$

${a}_{n} = 250 - 15 n + 15$

${a}_{n} = - 15 n + 265$