If 1/a, 1/b, 1/c, are in a.p prove that BC, ca, an are in a.p
hints: prove that ab- ca =ca- BC
Using 1/c - 1/b = 1/b - 1/a.

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Solution

To be in AP
1/c - 1/b = 1/b - 1/a.
(b-c)/bc= (a-b)/ab
ab-ca=ca-bc

This shows ca is the middle term
Hence ab,ca,bc are in AP.

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If 1/a, 1/b, 1/c, are in a.p prove that BC, ca, an are in a.p hints: prove that ab- ca =ca- BC Using 1/c - 1/b = 1/b - 1/a.

To be in AP 1/c - 1/b = 1/b - 1/a. (b-c)/bc= (a-b)/ab ab-ca=ca-bc This shows ca is the middle term Hence ab,ca,bc are in AP.

If 1/a, 1/b, 1/c, are in a.p prove that BC, ca, an are in a.p
hints: prove that ab- ca =ca- BC
Using 1/c - 1/b = 1/b - 1/a.

To be in AP
1/c - 1/b = 1/b - 1/a.
(b-c)/bc= (a-b)/ab
ab-ca=ca-bc

This shows ca is the middle term
Hence ab,ca,bc are in AP.