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 #719030 by stanbon(75887) $\"\"$ $\"About$

You can put this solution on YOUR website!
Use Mathematical Induction to show that the following statement is true for all natural numbers n: 1^3+2^3+3^3+...+ n^3 = n^2(n+1)^2/(4)
\n" ); document.write( "-------
\n" ); document.write( "Show it is true for n = 1::
\n" ); document.write( "1^3 = [1^2(1+1)^2]/4 = (1*2^2)/4
\n" ); document.write( "1 = 4/4
\n" ); document.write( "1 = 1
\n" ); document.write( "--------------
\n" ); document.write( "Assume it is true for n = k,
\n" ); document.write( " 1^3 + 2^3 + ... + k^3 = [k^2(k+1)^2]/4
\n" ); document.write( "------
\n" ); document.write( "Show it is true for n = k+1
\n" ); document.write( "1^3 + 2^3 + ..+ k^3 + (k+1)^3 = [k^2(k+1)^2]/4 + (k+1)^3
\n" ); document.write( "Factor to getl::
\n" ); document.write( "= (k+1)^2[k^2/4 + (k+1)]
\n" ); document.write( "= (k+1)^2[(k^2 + 4k+ 4)]/4
\n" ); document.write( "= (k+1)^2[((k+1)+1)^2]/4
\n" ); document.write( "----
\n" ); document.write( "So it is true for n = k+1.
\n" ); document.write( "Cheers,
\n" ); document.write( "Stan H.
\n" ); document.write( "----------- \n" ); document.write( "

\n" );