Q18 of 110 Page 41

If a, b, c are in A.P. then prove that are also in A.P.

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


Here, first term = a


Common difference, d1 = b – a … (1)


and d2 = c – b … (2)


Consider, and,


Common difference, d3 = -


=


= … (3)


⇒ d4 = -


=


= … (4)


From (1) and (2),


⇒ d1 = d2


⇒ b – a = c – b


Dividing both sides by abc,


⇒ =


⇒ d3 = d4 [From (3) and (4)]


Hence,, and are in A.P.


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