Need to find initial term and the common difference A.P

Question

What's given: Sum of the first 6 terms is equal to -12 and the sum of the LAST 8 terms of progression is equal to -224

I'm required to find the initial term and the common difference

Answer 1

Just let the first term as $a$, and common difference as $d$. Then general them is $a_n=a+(n-1)d$.

So sum of first 6 terms are $a_1+a_2+\cdots+a_6=a+(a+d)+\cdots+(a+5d)=6a+15d$, and it'll be -6. We can get $2a+5d=-2$.

And sum of 13th~20th terms are $a_{13}+a_{14}+\cdots+a_{20}=(a+12d)+(a+13d)+\cdots+(a+19d)=8a+124d$, and it'll be -224. We can get $2a+31d=-56$.

Then solve these, we can get $a=\frac{109}{26}, d=-\frac{27}{13}$.