Misc 16 - If a(1/b + 1/c), b(1/c + 1/a), c(1/a + 1/b) are AP - Arithmetic Progression (AP): Calculation based/Proofs

  1. Chapter 9 Class 11 Sequences and Series
  2. Serial order wise
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Misc 16 If a(1/(๐‘ ) " + " 1/๐‘), b(1/๐‘ " + " 1/๐‘Ž), c(1/๐‘Ž " + " 1/๐‘) are in AP., prove that a, b, c are in AP Given that a(1/(๐‘ ) " + " 1/๐‘), b(1/๐‘ " + " 1/๐‘Ž), c(1/๐‘Ž " + " 1/๐‘) are in AP. If a(1/(๐‘ ) " + " 1/๐‘), b(1/๐‘ " + " 1/๐‘Ž), c(1/๐‘Ž " + " 1/๐‘) are in AP Adding 1 to each term a(1/(๐‘ ) " + " 1/๐‘) + 1, b(1/๐‘ " + " 1/๐‘Ž) + 1, c(1/๐‘Ž " + " 1/๐‘) + 1 are in AP a(1/(๐‘ ) " + " 1/๐‘) + ๐‘Ž/๐‘Ž , b(1/๐‘ " + " 1/๐‘Ž) + ๐‘/๐‘ , c(1/๐‘Ž " + " 1/๐‘) + ๐‘/๐‘ are in AP a (1/(๐‘ ) " + " 1/๐‘ " + " 1/๐‘Ž) , b(1/๐‘ " + " 1/๐‘Ž " + " 1/(๐‘ )) , c(1/๐‘Ž " + " 1/(๐‘ ) " + " 1/๐‘) are in AP Divide each term by (1/๐‘Ž " + " 1/(๐‘ ) " + " 1/๐‘) ๐‘Ž(1/(๐‘ ) " + " 1/๐‘ " + " 1/๐‘Ž)/(1/๐‘Ž " + " 1/(๐‘ ) " + " 1/๐‘), ๐‘(1/๐‘ " + " 1/๐‘Ž " + " 1/(๐‘ ))/(1/๐‘Ž " + " 1/(๐‘ ) " + " 1/๐‘), ๐‘(1/๐‘Ž " + " 1/(๐‘ ) " + " 1/๐‘)/(1/๐‘Ž " + " 1/(๐‘ ) " + " 1/๐‘) are in AP a, b, c are AP Hence, a, b, c, are in AP Hence proved

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