1. Chapter 5 Class 12 Continuity and Differentiability
2. Serial order wise

Transcript

Misc 19 Using mathematical induction prove that ๐๏ทฎ๐๐ฅ๏ทฏ( ๐ฅ๏ทฎ๐๏ทฏ) = ๐๐ฅ๏ทฎ๐โ1๏ทฏ for all positive integers ๐. Let P ๐๏ทฏ : ๐๏ทฎ๐๐ฅ๏ทฏ ( ๐ฅ๏ทฎ๐๏ทฏ) = ๐๐ฅ๏ทฎ๐โ1๏ทฏ For ๐ = 1 LHS = ๐( ๐ฅ๏ทฎ1๏ทฏ) ๏ทฎ๐๐ฅ๏ทฏ = ๐๐ฅ๏ทฎ๐๐ฅ๏ทฏ = 1 โด LHS = RHS Thus ๐ ๐๏ทฏ is true for ๐ = 1 Let us assume that Let ๐ ๐๏ทฏ is true for ๐โ๐ต ๐ ๐๏ทฏ : ๐( ๐ฅ๏ทฎ๐๏ทฏ) ๏ทฎ๐๐ฅ๏ทฏ = ๐ ๐ฅ๏ทฎ๐โ1๏ทฏ Now We have to prove that P ๐+1๏ทฏ is true ๐ ๐+1๏ทฏ : ๐( ๐ฅ๏ทฎ๐ + 1๏ทฏ) ๏ทฎ๐๐ฅ๏ทฏ = ๐+1๏ทฏ ๐ฅ๏ทฎ๐ + 1 โ 1๏ทฏ ๐( ๐ฅ๏ทฎ๐ + 1๏ทฏ) ๏ทฎ๐๐ฅ๏ทฏ = ๐+1๏ทฏ ๐ฅ๏ทฎ๐๏ทฏ Taking L.H.S ๐( ๐ฅ๏ทฎ๐ + 1๏ทฏ) ๏ทฎ๐๐ฅ๏ทฏ = ๐( ๐ฅ๏ทฎ๐ ๏ทฏ. ๐ฅ ) ๏ทฎ๐๐ฅ๏ทฏ = ๐( ๐ฅ๏ทฎ๐๏ทฏ) ๏ทฎ๐๐ฅ๏ทฏ . ๐ฅ + ๐ ๐ฅ ๏ทฏ๏ทฎ๐๐ฅ๏ทฏ . ๐ฅ๏ทฎ๐ ๏ทฏ = ๐( ๐๏ทฎ๐๏ทฏ) ๏ทฎ๐๐๏ทฏ . ๐ฅ + 1 . ๐ฅ๏ทฎ๐ ๏ทฏ = ๐. ๐๏ทฎ๐โ๐๏ทฏ๏ทฏ . ๐ฅ+ ๐ฅ๏ทฎ๐๏ทฏ = ๐. ๐ฅ๏ทฎ๐โ1 + 1๏ทฏ .+ ๐ฅ๏ทฎ๐๏ทฏ = ๐. ๐ฅ๏ทฎ๐๏ทฏ+ ๐ฅ๏ทฎ๐๏ทฏ = ๐ฅ๏ทฎ๐๏ทฏ ๐+1๏ทฏ = R.H.S Hence proved Thus , ๐ ๐+1๏ทฏ is true when ๐ ๐๏ทฏ is true โด By principal of mathematical Induction ๐ ๐๏ทฏ : ๐๏ทฎ๐๐ฅ๏ทฏ ( ๐ฅ๏ทฎ๐๏ทฏ) = ๐๐ฅ๏ทฎ๐โ1๏ทฏ is true , ๐โ๐