Question 8 Exercise 4.2
Solutions of Question 8 of Exercise 4.2 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.
Question 8
If b+c−aa,c+a−bb,a+b−cc are in A.P, then prove 1a,1b,1c are in A.P.
Solution
Since
b+c−aa,c+a−bb,a+b−cc are in A.P, thus
∴c+a−bb−b+c−aa=a+b−cc−c+a−bbLetS=a+b+c2⇒a+b+c=2Sthen⇒a+b−c=2(S−c), a+c−b=2(S−b),andb+c−a=2(S−a)
then (1), becomes
2(S−b)b−2(S−a)a=2(S−c)c−2(S−b)b
Dividing both sides by 2
S−bb−S−aa=S−cc−S−bba(S−b)−b(S−a)ab=b(S−c)−c(S−b)bcaS−ab−bS+abab=bS−bc−cS+bcbc⇒(a−b)Sab=(b−c)Sbc
Dividing both sides by S
⇒a−bab=b−cbc⇒aab−bab=bbc−cbc⇒1b−1a=1c−1b⇒1a,1b,1c are in A.P.
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