document.write( "Question 723119: What is the sum of all the multiples of 4 through 4000? \n" ); document.write( "

Algebra.Com's Answer #443199 by KMST(5328) $\"\"$ $\"About$

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That is the sum of an arithmetic sequence with 1000 terms.
\n" ); document.write( "It is easy to see that the sum,
\n" ); document.write( "4+8+12+ .... +3992+3996+4000 is half of
\n" ); document.write( "(4+8+12+ .... +3992+3996+4000) + (4000+3996+3992+ .... +12+8+4) = (4+4000)+(8+3996)+(12+3992)+ ... +(3992+12)+(3996+8)+(4000+4)=4004+4004+4004+ ... +4004+4004+4004=1000*4004=4004000
\n" ); document.write( "So the sum is $\"highlight%282002000%29\"$ .
\n" ); document.write( "
\n" ); document.write( "We could write some fancy formulas, like
\n" ); document.write( " $\"a%5B1%5D=4\"$ , $\"d=4\"$ , $\"a%5Bk%5D%2B4=a%5Bk%2B1%5D\"$ ,
\n" ); document.write( " $\"a%5Bk%5D=a%5B1%5D%2Bd%28k-1%29\"$ and in this case $\"a%5Bk%5D=4%2B4%28k-1%29=4%2B4k-4=4k\"$
\n" ); document.write( "In this case, $\"n=1000\"$ $\"a%5Bn%5D=4%2A1000=4000\"$
\n" ); document.write( " $\"sum%28a%5Bk%5D%2Ck=1%2C1000%29=n%2A%28a%5B1%5D%2Ba%5Bn%5D%29%2F2\"$ and in this case
\n" ); document.write( " $\"sum%284k%2Ck=1%2C1000%29=1000%2A%284%2B4000%29%2F2=1000%2A4004%2F2=2002000\"$ \n" ); document.write( "

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