MR-383 / 8. problem

The fourth member of the arithmetic series is 7, and the fifth is -5.

Determine the first member of the series!

$${a}_{4}=7,{a}_{5}=-5$$

$${\left(}{1}{\right)}{a}_{4}={a}_{1}+3d=7$$

$${\left(}{2}{\right)}{a}_{5}={a}_{1}+4d=-5$$

$${\left(}{1}{\right)}{a}_{1}+3d=7$$

$${\left(}{2}{\right)}{a}_{1}+4d=-5$$

$${\left(}{2}{\right)}{-}{\left(}{1}{\right)}\left({a}_{1}+4d\right)-\left({a}_{1}+3d\right)=-5-7$$

$${\left(}{2}{\right)}{-}{\left(}{1}{\right)}{{a}}_{1}+4d{-}{{a}}_{1}-3d=-5-7$$

$$$$$$d=-12$$

$$$$$${\left(}{1}{\right)}{\to}{a}_{1}=7-3d$$

$$$$$${a}_{1}=7-3(-12)=7+36$$

$$$$$${a}_{1}=43$$

Write the 4th and 5th term of the sequence through the first term!

$${a}_{1}=43$$