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Prove-that-4n-lt-2-n-for-all-n-5-




Question Number 176102 by MathsFan last updated on 12/Sep/22
         Prove that        4n<2^n  for all n≥5
Provethat4n<2nforalln5
Answered by a.lgnaoui last updated on 12/Sep/22
n=5   2^5 =32         4×5=20<32  n=6   2^6 =64         4×6=24<64  n=7    2^7 =128      4×7=28<128  ...........................  n+1     2^(n+1)           4(n+1)=4n+4<2^n +4                                     n[4(n+1)]<n(2^n +4)<n(2×2^n )<n2^(n+1)   donc   4(n+1)<2^(n+1)   n est vrai pour tout n≥5 (n∈ N)
n=525=324×5=20<32n=626=644×6=24<64n=727=1284×7=28<128n+12n+14(n+1)=4n+4<2n+4n[4(n+1)]<n(2n+4)<n(2×2n)<n2n+1donc4(n+1)<2n+1nestvraipourtoutn5(nN)
Commented by MathsFan last updated on 13/Sep/22
  Merci Monsieur
Merci Monsieur
Answered by mahdipoor last updated on 12/Sep/22
if 4a<2^a  ⇒  4a+4<2^a +4=2^a ×(1+(4/2^a ))<2^a ×2   (a>5)  ⇒4(a+1)<2^(a+1)     and 4×5<2^5   ⇒⇒ 4n<2^n  for ∀n≥5
if4a<2a4a+4<2a+4=2a×(1+42a)<2a×2(a>5)4(a+1)<2a+1and4×5<25⇒⇒4n<2nforn5
Commented by MathsFan last updated on 13/Sep/22
thank you sir
thankyousir

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