If a, b, c are in AP, show that

(i) (b + c – a), (c + a – b), (a + b – c) are in AP.


(ii) (bc – a2), (ca – b2), (ab – c2) are in AP.


(i) (b + c – a), (c + a – b), (a + b – c) are in AP.


To prove: (b + c – a), (c + a – b), (a + b – c) are in AP.


Given: a, b, c are in A.P.


Proof: Let d be the common difference for the A.P. a,b,c


Since a, b, c are in A.P.


b – a = c – b = common differnce


a – b = b – c = d


2(a – b) = 2(b – c) = 2d … (i)


Considering series (b + c – a), (c + a – b), (a + b – c)


For numbers to be in A.P. there must be a common difference between them


Taking (b + c – a) and (c + a – b)


Common Difference = (c + a – b) - (b + c – a)


= c + a – b – b – c + a


= 2a – 2b


= 2(a – b)


= 2d [from eqn. (i)]


Taking (c + a – b) and (a + b – c)


Common Difference = (a + b – c) - (c + a – b)


= a + b – c – c – a + b


= 2b – 2c


= 2(b – c)


= 2d [from eqn. (i)]


Here we can see that we have obtained a common difference between numbers i.e. 2d


Hence, (b + c – a), (c + a – b), (a + b – c) are in AP.


(ii) (bc – a2), (ca – b2), (ab – c2) are in AP.


To prove: (bc – a2), (ca – b2), (ab – c2) are in AP.


Given: a, b, c are in A.P.


Proof: Let d be the common difference for the A.P. a,b,c


Since a, b, c are in A.P.


b – a = c – b = common differnce


a – b = b – c = d … (i)


Considering series (bc – a2), (ca – b2), (ab – c2)


For numbers to be in A.P. there must be a common difference between them


Taking (bc – a2) and (ca – b2)


Common Difference = (ca – b2) – (bc – a2)


= [ca – b2 – bc + a2]


= [ca – bc + a2 – b2]


= [c (a – b) + (a + b) (a – b)]


= [(a – b ) (a + b + c)]


a – b = d, from eqn. (i)


[(d) (a + b + c)]


Taking (ca – b2) and (ab – c2)


Common Difference = (ab – c2) – (ca – b2)


= [ab – c2 – ca + b2]


= [ab – ca + b2 – c2]


= [a (b – c) + (b – c) (b + c)]


= [(b – c) (a + b + c)]


b – c = d, from eqn. (i)


[(d) (a + b + c)]


Here we can see that we have obtained a common difference between numbers i.e. [(d) (a + b + c)]


Hence, (bc – a2), (ca – b2), (ab – c2) are in AP.


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