SOLUTION: If a,b,c are in A.P., show that (i) 2(a-b) = a-c = 2(b-c) (ii)(a-c)^2 = 4(b^2-ac)

Algebra ->  Sequences-and-series -> SOLUTION: If a,b,c are in A.P., show that (i) 2(a-b) = a-c = 2(b-c) (ii)(a-c)^2 = 4(b^2-ac)      Log On


   

Question 1090199: If a,b,c are in A.P., show that
(i) 2(a-b) = a-c = 2(b-c)
(ii)(a-c)^2 = 4(b^2-ac)
Found 2 solutions by htmentor, ikleyn:
Answer by htmentor(1343) About Me  (Show Source):
You can put this solution on YOUR website!
It is not stated whether the terms are consecutive, but we will proceed under that assumption.
A general expression for the term a is a = a1 + (n-1)d, assuming a is the n-th term, and d is the common difference
So b = a + d = and c = a + 2d
(i)
2(a - b) = -2d
a - c = -2d
2(b - c) = -2d
(ii)
(a - c)^2 = 4d^2
4(b^2 - ac) = 4( (a+d)^2 - a(a+2d) ) = 4( a^2 + 2ad + d^2 - a^2 - 2ad ) = 4d^2

Answer by ikleyn(52903) About Me  (Show Source):
You can put this solution on YOUR website!
.
In connection to this problem, see the lesson
    - One characteristic property of arithmetic progressions
in this site.

The characteristic property of any Arithmetic progression is THIS: 
    for any three consecutive terms  a, b and c  the middle term is equal to the half-sum of its neighbors.

There is a bunch of lessons on arithmetic progressions in this site:
    - Arithmetic progressions
    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
    - Mathematical induction and arithmetic progressions
    - One characteristic property of arithmetic progressions
    - Solved problems on arithmetic progressions


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".