# What is the first term of an arithmetic sequence in which the seventh term is 3 and the sixteenth term is -15?

Jul 17, 2015

The first term of the sequence is $\textcolor{red}{15}$.

#### Explanation:

We know that ${a}_{7} = 3$ and ${a}_{16} = - 15$.

We also know that ${a}_{n} = {a}_{1} + \left(n - 1\right) d$.

${a}_{16} = {a}_{1} + \left(16 - 1\right) d$

Equation (1): $- 15 = {a}_{1} + 15 d$

and

${a}_{7} = {a}_{1} + \left(7 - 1\right) d$

Equation (2): $3 = {a}_{1} + 6 d$

Subtract Equation (2) from Equation (1).

$- 15 - 3 = 15 d - 6 d$

$- 18 = 9 d$

$d = - \frac{18}{9}$

Equation (3): $d = - 2$

Substitute Equation (3) in Equation (1).

$- 15 = {a}_{1} + 15 d$

$- 15 = {a}_{1} + 15 \left(- 2\right) = {a}_{1} - 30$

${a}_{1} = 30 - 15$

${a}_{1} = 15$