.
\n" );
document.write( "Please solve:
\n" );
document.write( "Sn=a^2+(a+d)^2+(a+2d)^2+(a+3d)^2+....+[(a+(n-1)d]^2
\n" );
document.write( "Sn=a²+(a+d)²+(a+2d)²+(a+3d)²+....+[(a+(n-1)d]²
\n" );
document.write( "----------------------------------------------------\r
\n" );
document.write( "\n" );
document.write( "\r\n" );
document.write( "Let us consider one typical term 
. It is\r\n" );
document.write( "
+ 
+ 
.\r\n" );
document.write( "\r\n" );
document.write( "We need to sum up n such terms/trinomials from k = 0 to k = n-1.\r\n" );
document.write( "\r\n" );
document.write( "By combining the first addends of these trinomials, you will get 
, right?\r\n" );
document.write( "\r\n" );
document.write( "By combining the second addends of these trinomials, you will get 
.\r\n" );
document.write( "\r\n" );
document.write( "If you know it, this sum 
is equal to 
. \r\n" );
document.write( "\r\n" );
document.write( "It is the sum of a special arithmetic progression which is the sequence of the first (n-1) natural numbers. \r\n" );
document.write( "If you don't know it, see the lesson Arithmetic progressions in this site. \r\n" );
document.write( "\r\n" );
document.write( "By combining the third addends of these trinomials, you will get 
.\r\n" );
document.write( "\r\n" );
document.write( "This sum of squares of the first (n-1) natural numbers 
is equal to 
.\r\n" );
document.write( "\r\n" );
document.write( "For the proof see, for example, the lesson Mathematical induction for sequences other than arithmetic or geometric in this site. \r\n" );
document.write( "\r\n" );
document.write( "Now we can finalize our calculations.\r\n" );
document.write( "\r\n" );
document.write( "
= 
= \r\n" );
document.write( "\r\n" );
document.write( "= + 
.
+ 
= \r\n" );
document.write( "\r\n" );
document.write( "= + 
+ 
.
.\r\n" );
document.write( "
\n" );
document.write( " \n" );
document.write( "![\"\"](\"/images/stars/stars-13.gif\")
