-6

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

guy
  • 1
  • How many terms does the total sequence have? I can't solve it without this information. Afterall... what did you have tried? – MH.Lee Oct 25 '21 at 18:38
  • 1
    Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Shaun Oct 25 '21 at 18:43
  • @Nightflight, total sequence has 20 terms. I did try to solve with this method (answer by Maung Maung Sein )but I'm stuck with the second part of this question. – guy Oct 25 '21 at 18:50
  • 1
    guy, please add that information in an edit to your post, and not in comments. The links provided by Shaun will help give you additional tips, because as it stands, your post is a statement of a question, and implies "solve it for me." – amWhy Oct 25 '21 at 18:58
  • 1
    possible duplicate: https://math.stackexchange.com/questions/2782042/arithmetic-progression-ap – Matt Groff Oct 25 '21 at 18:59

1 Answers1

0

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 $a=\frac{109}{26}, d=-\frac{27}{13}$.

MH.Lee
  • 5,568
  • Thank you sir, It makes sense know. The solution was right under my nose. – guy Oct 25 '21 at 19:48
  • The sum of the first six terms is $ \ -12 \ \ , $ so your first system equation should read " $ \ 2a + 5d \ = \ \mathbf{-4} \ \ . $ You will obtain $ \ d \ = \ -2 \ \ , \ \ a \ = \ 3 \ \ , $ a somewhat nicer set of numbers. –  Feb 10 '22 at 04:40