If a,b,c are in Ap . Then prove that ab+bc+ca are in Ap
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We have to prove that bc, ca, and ab are in HP, which is another way of saying that 1/bc, 1/ca, and 1/ab are in AP. This simply means we have to prove that
2(1/ca) = 1/ab + 1/bc
=> 2/ca = 1/ab + 1/bc
We are given that a, b, and c are in AP, which means
2b = a + c
Diving the entire expression by abc, we'll have
=> 2b/abc = a/abc + c/abc
=> 2/ca = 1/bc + 1/ab
which means 1/bc, 1/ca, and 1/ab are in AP, or bc, ca and ab are in AP
2(1/ca) = 1/ab + 1/bc
=> 2/ca = 1/ab + 1/bc
We are given that a, b, and c are in AP, which means
2b = a + c
Diving the entire expression by abc, we'll have
=> 2b/abc = a/abc + c/abc
=> 2/ca = 1/bc + 1/ab
which means 1/bc, 1/ca, and 1/ab are in AP, or bc, ca and ab are in AP
Bhardwajsingh:
Thank you
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