document.write( "Question 1104281: Use Mathematical Induction to show that the following statement is true for all natural numbers n: $\"+1%5E3%2B2%5E3%2B3%5E3%2B...%2B\"$ $\"n%5E3+=+n%5E2%28n%2B1%29%5E2%2F%284%29+\"$
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #719032 by math_helper(2461) $\"\"$ $\"About$

You can put this solution on YOUR website!
For n=2:
\n" ); document.write( " $\"+1%5E3+%2B+2%5E3+=+9++\"$
\n" ); document.write( "and
\n" ); document.write( " $\"+%28%282%5E2%2A%282%2B1%29%5E2%29+%2F+4%29+=+%284%2A9%29%2F4+=+9++\"$
\n" ); document.write( "—
\n" ); document.write( "For n=3:
\n" ); document.write( " $\"+1%5E3+%2B+2%5E3+%2B+3%5E3+=+1%2B8%2B27+=+36+\"$
\n" ); document.write( "and
\n" ); document.write( " $\"++%28%283%5E2%2A%283%2B1%29%5E2%29+%2F+4%29+=+%289%2A16%29%2F4+=+36+\"$
\n" ); document.write( "—
\n" ); document.write( "Assume it is true for n=k. That is $\"+1%5E3%2B2%5E3+\"$ + ... + $\"+%28k-1%29%5E3+%2B+k%5E3++=+%28k%5E2%28k%2B1%29%5E2%29+%2F+4+\"$ (*)
\n" ); document.write( "—
\n" ); document.write( "Now let n=k+1:
\n" ); document.write( " $\"+1%5E3+%2B+2%5E3+%2B+3%5E3+\"$ + ... + $\"+k%5E3+%2B+%28k%2B1%29%5E3+\"$
\n" ); document.write( "Which, upon substituting (*), gets us to:
\n" ); document.write( " = $\"+%28k%5E2%28k%2B1%29%5E2%29%2F4+%2B+%28k%2B1%29%5E3+\"$
\n" ); document.write( "factoring:
\n" ); document.write( "= $\"+%28%28k%2B1%29%5E2%29%2F4+%2A+%28k%5E2+%2B+%284k%2B+4%29%29+\"$
\n" ); document.write( "= $\"+%28%28k%2B1%29%5E2%29%2F4+%2A+%28k%2B2%29%5E2+\"$ \r
\n" ); document.write( "\n" ); document.write( "QED.
\n" ); document.write( "(Assuming (*) is true for n=k leads to (*) being true for n=k+1)
\n" ); document.write( " \n" ); document.write( "

\n" );