Question

If a b, c are in AP show that (b+c-a), (c+a-b), (a + b -c) are in AP.
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Answer 1

Note :

★ A.P. (Arithmetic Progression) : A sequence in which the difference between the consecutive terms are equal is said to be in A.P.

★ If a1 , a2 , a3 , . . . , an are in AP , then

a2 - a1 = a3 - a2 = a4 - a3 = . . .

★ If each terms of an AP is multiplied or divided by same quantity , then the resulting sequence is an AP .

★ If same quantity is added or subtracted in each term of an AP then the resulting sequence is an AP .

Solution :

→ Given : a , b , c are in AP

→ To prove :

(b + c - a) , (c + a - b) , (a + b - c) are in AP

Proof :

We have ,

→ a , b , c are in AP

Now ,

Multiplying the AP by -2 , we get ;

→ -2a , -2b , -2c are in AP

Now ,

Adding (a+b+c) in the AP , we get ;

→ (a+b+c)-2a , (a+b+c)-2b , (a+b+c)-2c are in AP

→ (b+c-a) , (c+a-b) , (a+b-c) are in AP

Hence Proved .