If a,b,c are in AP Prove that a(1/b+1/c) , b(1/c+1/a) , c(1/a+1/b) are also in AP
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Given 1a, 1/b, 1/c are in A.P
Prove that a(b+c), b(c+a),c(a+b) .... are in AP
Solution:
ab+ac, bc+ab, ca+cb are in A.p
∵t3-t2=t2-t1
(bc+ab)-(ab+ac)=(ac+bc)-(bc+ab)
bc+ab-ab-ac=ac+bc-bc-ab
bc-ac=ac-ab
2ac=bc+ab
divide the equation on both sides by abc, we get
2/b=1/a+1/c
this is the condition for A.p
hence proved
Prove that a(b+c), b(c+a),c(a+b) .... are in AP
Solution:
ab+ac, bc+ab, ca+cb are in A.p
∵t3-t2=t2-t1
(bc+ab)-(ab+ac)=(ac+bc)-(bc+ab)
bc+ab-ab-ac=ac+bc-bc-ab
bc-ac=ac-ab
2ac=bc+ab
divide the equation on both sides by abc, we get
2/b=1/a+1/c
this is the condition for A.p
hence proved
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