# Math Help - series and sequences question

1. ## series and sequences question

The sum of the first two terms of an arithmetic series is 47.
The thirtieth term of this series is -62. Find the first term of the series and the common difference.
I'm stuck on the first bit. I'm not quite sure how to work out which two terms equal the sum of 47....thanks.

2. Hello,
Originally Posted by girlpower1991
The sum of the first two terms of an arithmetic series is 47.
The thirtieth term of this series is -62. Find the first term of the series and the common difference.
I'm stuck on the first bit. I'm not quite sure how to work out which two terms equal the sum of 47....thanks.
Let $\delta$ be the common difference and $a_1, ~a_2,~\text{etc}$ the terms of the series.

$a_2=a_1+ \delta$

But we know that $a_1+a_2=47$

Thus $2a_1+ \delta=47$

The thirtieth term is $-62=a_{30}=a_1+29 \delta$

So we have the system :

$\left\{\begin{array}{ll} 2a_1+\delta=47 \quad (1) \\ a_1+29 \delta=-62 \quad (2) \end{array} \right.$

$2*(2)-(1) ~:~ 2(a_1+29 \delta)-(2a_1+\delta)=2*(-62)-47$

The $a_1$ simplify and we get :
$58 \delta- \delta=-124-47$

$57 \delta=-171$

$\boxed{\delta=-3}$