document.write( "Question 1051728: Prove by mathematical induction
\n" ); document.write( "1^2+3^2+5^2+.......upto n terms = n(4n^2-1)/3 \n" ); document.write( "

Algebra.Com's Answer #667192 by rothauserc(4718) $\"\"$ $\"About$

You can put this solution on YOUR website!
1) 1^2+3^2+5^2+.......up to (2n-1)^2 = n(4n^2-1)/3 = n(2n-1)(2n+1)/3
\n" ); document.write( ":
\n" ); document.write( "show that 1) is true for n = 1, that is
\n" ); document.write( ":
\n" ); document.write( "1^2 = 1(3) / 3 = 1
\n" ); document.write( ":
\n" ); document.write( "now assume 1) is true for n, then show it is true for n+1
\n" ); document.write( ":
\n" ); document.write( "n(2n-1)(2n+1)/3 + (2n+1)^2 = (2n+1)(2n^2 -n +6n +3) / 3 =
\n" ); document.write( ":
\n" ); document.write( "*********************************************************
\n" ); document.write( "(2n+1)(2n^2 +5n +3) /3 = (2n+1)(2n+3)(n+1) / 3
\n" ); document.write( ":
\n" ); document.write( "note that if we substitute (n+1) for n in 1) we get
\n" ); document.write( ":
\n" ); document.write( "(n+1)(2(n+1)-1)(2(n+1)+1)/3 = (n+1)(2n+1)(2n+3)/3
\n" ); document.write( "*********************************************************
\n" ); document.write( ":
\n" ); document.write( "the statement has been proven with induction
\n" ); document.write( ": \n" ); document.write( "

\n" );