Question 1181280: The first four terms of an arithmetic sequence are 2, a-b, 2a+b+7, a-3b . Find a and b.
Found 5 solutions by josgarithmetic, MathLover1, MathTherapy, greenestamps, ikleyn:
Answer by josgarithmetic(39797)   (Show Source):
Answer by MathLover1(20855)  (Show Source):
You can put this solution on YOUR website!
The first four terms of an arithmetic sequence are
 ,  ,  , 
=> first term is
in an arithmetic sequence we add common difference  to get next term, so
 ....eq.1
add common difference to second term to get third term
 .......eq.2
add common difference to third term to get fourth term
 ..........eq.3
solve the system
....eq.1
.......eq.2
..........eq.3
----------------------------------------
....eq.1, solve for
 ........eq.1a
.......eq.2, solve for
 ................eq.2a
from eq.1a and eq.2a we have
 ............., solve for 
go to
........eq.1a, substitute
 ......................eq.1b
go to
..........eq.3, substitute and
go to
......................eq.1b, substitute 
answer: , , common difference 
and your terms are:
,  ,  , 
,  ,  , 
Answer by MathTherapy(10809)  (Show Source):
Answer by greenestamps(13334)  (Show Source):
You can put this solution on YOUR website!
You have received, to this point, three responses from different tutors, all showing either valid complete algebraic solutions, or at least a way to set up the problem for solving.
All of those responses use the same approach: the difference between the first and second terms is the same as the difference between the second and third terms, and it is the same as the difference between the third and fourth terms.
Many times it is worthwhile to take a close look at the given information to see if there is a quick path to the answer, before plunging into the algebra.
There is an easier path to the solution to this particular problem.
The second term is a-b; the fourth term is a-3b. The difference between the second and fourth terms, -2b, is twice the common difference -- so the common difference is -b.
The first term is 2, and the second term is a-b. Since we know the common difference is -b, a is 2.
To find b, we know adding the common difference to the second term will give the third term:
ANSWER: a=2; b=-3
Answer by ikleyn(53762)  (Show Source):
You can put this solution on YOUR website! .
In this problem, there are two unknowns, "a" and "b";
so, you need to construct two equations for these two unknowns.
There is the STANDARD WAY to construct these two equations:
first equation says that  =  (both sides represent the common difference)
second equation says that =  (both sides represent the common difference, again).
It is the way on how @MathTherapy solves the problem in his post.
How @Mathlover1 and @josgarithmetic do it, it looks like chewing gum in the mouth
instead of solving problem and instead of right teaching . . .
You, probably, will solve tens of such problems in your life (in your school years).
THEREFORE, you need to know the right way and do not spend your time for nothing . . .
To see many other similar (and different) problems solved, look into the lessons
- One characteristic property of arithmetic progressions
- Finding number of terms of an arithmetic progression
- Inserting arithmetic means between given numbers
in this site.
Also, you have this free of charge online textbook in ALGEBRA-II in this site
- ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this online textbook under the topic
"Arithmetic progressions".
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