Last updated at May 6, 2021 by Teachoo
Transcript
Misc 19 Using mathematical induction prove that ๐/๐๐ฅ(๐ฅ^๐) = ใ๐๐ฅใ^(๐โ1) for all positive integers ๐. Let ๐(๐) : ๐/๐๐ฅ (๐ฅ^๐) = ใ๐๐ฅใ^(๐โ1) For ๐ = ๐ Solving LHS (๐(๐ฅ^1)" " )/๐๐ฅ = ๐๐ฅ/๐๐ฅ = 1 = RHS Thus, ๐ท(๐) is true for ๐ = 1 Let us assume that ๐ท(๐) is true for ๐โ๐ต ๐ท(๐) : (๐ (๐ฅ^๐))/๐๐ฅ = ใ๐ ๐ฅใ^(๐โ1) Now We have to prove that P(๐+๐) is true ๐(๐+1) : (๐(๐ฅ^(๐ + 1))" " )/๐๐ฅ = ใ(๐+1) ๐ฅใ^(๐ + 1 โ 1) (๐(๐ฅ^(๐ + 1)))/๐๐ฅ = ใ(๐+1) ๐ฅใ^๐ Taking L.H.S (๐(๐ฅ^(๐ + 1)))/๐๐ฅ = (๐(๐ฅ^(๐ ). ๐ฅ))/๐๐ฅ Using product rule As (๐ข๐ฃ)โ = ๐ขโ๐ฃ + ๐ฃโ๐ข where u = xk & v = x = (๐(๐ฅ^๐)" " )/๐๐ฅ . ๐ฅ + ๐(๐ฅ )/๐๐ฅ . ๐ฅ^(๐ ) = (๐ (๐^๐)" " )/๐ ๐ . ๐ฅ + 1 . ๐ฅ^(๐ ) = (ใ๐. ๐ใ^(๐โ๐) ) . ๐ฅ+๐ฅ^๐ = ใ๐. ๐ฅใ^(๐โ1 + 1) .+๐ฅ^๐ = ใ๐. ๐ฅใ^๐+๐ฅ^๐ = ๐ฅ^๐ (๐+1) = R.H.S Hence proved (From (1): (๐(๐ฅ^๐ ") " )/๐๐ฅ = ใ๐ ๐ฅใ^(๐โ1) ) Thus , ๐ท(๐+๐) is true when ๐ท(๐) is true Therefore, By Principle of Mathematical Induction ๐(๐) : ๐/๐๐ฅ (๐ฅ^๐) = ใ๐๐ฅใ^(๐โ1) is true for all ๐โ๐ต
Proofs
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