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a^n -1=(a-1)(a^(n-1) + a^(n-2) +.......+ a+1)
\n" ); document.write( "We need to prove that it is true for n=1 or n=2,and
\n" ); document.write( "we need to prove that if it is true for n=k, it must be true for n=k+1.
\n" ); document.write( "is easy to prove for n=1 or for n=2
\n" ); document.write( "For n=1, a^(n-1)=a^(1-1)=a^0=1, and the long sum in parenthesis,
\n" ); document.write( "going from a^0=1 down to a^0=1, is just 1.
\n" ); document.write( "Substituting n=1 in a^n -1=(a-1)(a^(n-1) + a^(n-2) +.......+ a+1), we get
\n" ); document.write( "a^1 - 1=(a-1)(1), which is trivial.
\n" ); document.write( "For n=2, substituting we get
\n" ); document.write( "a^2 - 1=(a-1)(a+1) which we know is true. (It's one of those special products we learn for factoring).
\n" ); document.write( "If we assume that a^n - 1=(a-1)(a^(n-1) + a^(n-2) +.......+ a+1) is true for any positive integer
\n" ); document.write( "a^k - 1=(a-1)(a^(k-1) + a^(k-2) +.......+ a+1)
\n" ); document.write( "Adding 1 to both sides, we get
\n" ); document.write( "a^k =(a-1)(a^(k-1) + a^(k-2) +.......+ a+1)+1
\n" ); document.write( "Multiplying both sides times a, we get
\n" ); document.write( "a*a^k =((a-1)(a^(k-1) + a^(k-2) +.......+ a+1)+1)a
\n" ); document.write( "Distributing
\n" ); document.write( "a^(k+1) =(a-1)(a^(k-1) + a^(k-2) +.......+ a+1)*a+1*a
\n" ); document.write( "Applying the associative property
\n" ); document.write( "a^(k+1) =(a-1)((a^(k-1) + a^(k-2) +.......+ a+1)*a}+1*a
\n" ); document.write( "Distributing
\n" ); document.write( "a^(k+1) =(a-1)(a^(k-1+1) + a^(k-2+1) +.......+ a*a+1*a)+a
\n" ); document.write( "a^(k+1) =(a-1)(a^k + a^(k-1) +.......+ a^2+a)+a
\n" ); document.write( "a^(k+1)-1 =(a-1)(a^k + a^(k-1) +.......+ a^2+a)+a-1
\n" ); document.write( "a^(k+1)-1 =(a-1)(a^k + a^(k-1) +.......+ a^2+a)+(a-1)
\n" ); document.write( "a^(k+1)-1 =(a-1)((a^k + a^(k-1) +.......+ a^2+a)+1)
\n" ); document.write( "a^(k+1)-1 =(a-1)(a^k + a^(k-1) +.......+ a^2+a+1) which is
\n" ); document.write( "a^n -1=(a-1)(a^(n-1) + a^(n-2) +.......+ a+1) for n=k+1 \n" ); document.write( "