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If a,b,c are in ap then show that1/ba, 1/ca, 1/ab are in ap
If a,b,c are in ap, thentheir common difference would be:b – a and c-band therfore,2b = a+c..................(1)If 1/ba, 1/ca, 1/cb are in A.P,then2/ca = 1/ba + 1/cbtaking L.C.M abc and calculating we get,again2b = a + c.................(2)hence eqn. (1) and (2) are similar therefore this series is also in A.P.
If a,b,c are in AP, then c,b,a are in AP which means c/abc, b/abc, a/abc are in AP which gives 1/ab, 1/bc, 1ca are in AP.
swers are the finest of the finest.If a,b,c are in AP,b-a = c-bto prove 1/bc, 1/ca, 1/ab are in AP, we need to show 1/ca - 1/bc = 1/ab - 1/caLHS:1/ca - 1/bc = (b-a)/abc RHS:1/ab - 1/ca = (c-b)/abcbut we know that (b-a) = (c-b)thus LHS = RHS.Hence 1/bc, 1/ca, 1/ab are in AP.Note: Calculation of 1/ca - 1/bcYou need to first take the LCM of ca and bc which is abc and do the calculation. I have calculated like this above.Or\frac{1}{ca} - \frac{1}{bc} = \frac{bc-ca}{abc^{2}} = \frac{c(b-a))}{abc^{2}} = \frac{b-a}{abc} Here instead of taking LCM, i have multiplied the terms, which is fine.
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