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)

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)   (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(52905)   (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".

RELATED QUESTIONS
If a,b,c, are in harmonic progression shown that (1/a+1/b-1/c)(1/b+1/c... (answered by Edwin McCravy)
Given that p(A)= 1/4 p(B)= 1/5 p(C)= 1/2 Compute the following 1)... (answered by MathLover1)
If a:b :: c:d, show that: ac : bd =... (answered by robertb)
Simplify {{{ (a(b+c) + (b+c)^2)/(b^2-c^2 + ab-ac)... (answered by math_helper,MathLover1)
If three numbers a,b,c are in arithmetic progression then: a^2(b+c),b^2(c+a) and... (answered by CubeyThePenguin)
If three numbers a,b,c are in arithmetic progression then: a^2(b+c),b^2(c+a) and... (answered by CubeyThePenguin)
If a^2,b^2,c^2 are in A.P,show that b+c,c+a,a+b are in... (answered by ikleyn)
If a,b,c are in AP then prove that a^2(b+c),b^2(c+a),c^2(a+b) are in... (answered by Edwin McCravy)
Q1 If a,b,c,d are in Harmonic progression show that(a-c)(b-d)=4(a-b)(c-d) Q2 If... (answered by palanisamy)