document.write( "Question 935544: use mathematical induction to prove the summation of 
\n" ); document.write( "i=1
\n" ); document.write( "4i-1=n(2n+1) \n" ); document.write( "

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\n" ); document.write( " $\"sum%28%284i-1%29%2C+i=1%2C+i=n+%29=n%282n%2B1%29\"$
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\n" ); document.write( "For $\"n=1\"$ , what we need to prove is
\n" ); document.write( " $\"sum%28%284i-1%29%2C+i=1%2C+i=1+%29=1%282%2A1%2B1%29\"$
\n" ); document.write( " $\"sum%28%284i-1%29%2C+i=1%2C+i=1+%29=4%2A1-1=4-1=3\"$ is the \"sum\" of just the first term, $\"3\"$ .
\n" ); document.write( " $\"1%282%2A1%2B1%29=1%282%2B1%29=1%2A3=3\"$ has the same value.
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\n" ); document.write( "After that we have to prove that
\n" ); document.write( "if $\"sum%28%284i-1%29%2C+i=1%2C+i=n+%29=n%282n%2B1%29\"$ is true for $\"n=k\"$ ,
\n" ); document.write( "it must be true for $\"n=k%2B1\"$ .
\n" ); document.write( "For $\"n=k\"$ ,
\n" ); document.write( "we say it is true that $\"sum%28%284i-1%29%2C+i=1%2C+i=n+%29=n%282n%2B1%29\"$ , so
\n" ); document.write( " $\"sum%28%284i-1%29%2C+i=1%2C+i=k+%29=k%282k%2B1%29\"$ .
\n" ); document.write( "Then, we must prove that it is true for $\"n=k%2B1\"$ , meaning that
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