document.write( "Question 338207: Use induction to show
\n" ); document.write( "1+2n_<3^n for n_>1\r
\n" ); document.write( "\n" ); document.write( "I'm really having a hard time understanding this process . I think if I see some examples it will help me solve it on my own \n" ); document.write( "
Algebra.Com's Answer #242417 by stanbon(75887)\"\" \"About 
You can put this solution on YOUR website!
1+2n_<3^n for n_>1
\n" ); document.write( "---------------------------------\r
\n" ); document.write( "\n" ); document.write( "You show it is true for n = 1.
\n" ); document.write( "Then you assume it is true for n = k
\n" ); document.write( "and prove that is is true for n = k+1.
\n" ); document.write( "----
\n" ); document.write( "If you can do that, you know it is true
\n" ); document.write( "for 1+1 = 2, 2+1=3, 3+1=4, etc.
\n" ); document.write( "==============
\n" ); document.write( "Your Problem
\n" ); document.write( "True for n = 1 ?
\n" ); document.write( "1+2*1 <= 3^1
\n" ); document.write( "1+2 <= 3
\n" ); document.write( "3<=3
\n" ); document.write( "True
\n" ); document.write( "------------------
\n" ); document.write( "Assume true for n = k
\n" ); document.write( "1+2k <= 3^k
\n" ); document.write( "Note: We know it is true for k=1 because we did that above.
\n" ); document.write( "-------------------
\n" ); document.write( "Prove it is true for n = k+1
\n" ); document.write( "1+2(k+1) = (1+2k)+2
\n" ); document.write( "substitute 3^k for 1+2k to get
\n" ); document.write( "<= 3^k+2 <= 3^k+3 = 3^k*3 = 3^(k+1)
\n" ); document.write( "So 1+2(k+1) <= 3^(k+1)
\n" ); document.write( "--------
\n" ); document.write( "And that is what you were trying to prove
\n" ); document.write( "--------
\n" ); document.write( "So, 1+2n <= 3^n for all n>=1
\n" ); document.write( "=================================
\n" ); document.write( "Cheers,
\n" ); document.write( "Stan H.
\n" ); document.write( " \n" ); document.write( "
\n" );