# How do you find the first term and common difference if The 6th term of the sequence is -2, The 14th term of the sequence is -18?

Jun 10, 2016

First term: $8$
Common difference: $\left(- 2\right)$

#### Explanation:

For an arithmetic sequence, ${a}_{i}$, with a common difference of $d$
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{{a}_{k + n} = {a}_{k} + n \cdot d}$

We are told
$\textcolor{w h i t e}{\text{XXX")a_14=-18color(white)("XX")andcolor(white)("XX}} {a}_{6} = - 2$

Using $k = 6$ we have $n = 8$
$\textcolor{w h i t e}{\text{XXX}} {a}_{14} = \textcolor{red}{{a}_{6 + 8} = {a}_{6} + 8 \cdot d}$

$\textcolor{w h i t e}{\text{XXX}} - 18 = - 2 + 8 d$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow d = - 2$

Now setting $k = 1$
$\textcolor{w h i t e}{\text{XXX}} {a}_{6} = \textcolor{red}{{a}_{1 + 5} = {a}_{1} + 5 \times \left(- 2\right)}$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow - 2 = {a}_{1} + \left(- 10\right)$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow {a}_{1} = 8$