document.write( "Question 957846: How can we prove the equation below?
\n" ); document.write( "1*1!+2*2!+3*3!+...+n*n!=(n+1)!-1 \n" ); document.write( "

Algebra.Com's Answer #585343 by amarjeeth123(569) $\"\"$ $\"About$

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We can prove this equation using mathematical induction.
\n" ); document.write( "Let n=1
\n" ); document.write( "LHS=1*1!=1
\n" ); document.write( "RHS=2!-1=2-1=1
\n" ); document.write( "LHS=RHS
\n" ); document.write( "The equation holds true for n=1
\n" ); document.write( "We will assume that the equation holds true for n=k
\n" ); document.write( "1*1!+2*2!+.........+k*k!=(k+1)!-1......equation 1
\n" ); document.write( "For n=k+1 we have
\n" ); document.write( "LHS=1*1!+2*2!+.........+k*k!+(k+1)*(k+1)!
\n" ); document.write( " =(k+1)!-1+(k+1)*(k+1)!
\n" ); document.write( " =(k+1)!(k+1+1)-1
\n" ); document.write( " =(k+2)(k+1)!-1
\n" ); document.write( " =RHS
\n" ); document.write( "Therefore the equation holds true for all N. \n" ); document.write( "

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