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document.write( "18 \r
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document.write( "(a) Find 1 + 2 + . . . + 24.\r\n" );
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document.write( "(b) Show that 1/n, 2/n + ... n/n = (n+1)/2\r\n" );
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document.write( "(c) Hence find the sum of the first 300 terms of\r\n" );
document.write( " 1/1 + 1/2 + 2/2 + 1/3 + 2/3 + 3/3 + 1/4 + 2/4 + 3/4 +4/4 + ...\r\n" );
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document.write( "Well known fact is that the sum of the first n natural numbers\r\n" );
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document.write( "1 + 2 + 3 + . . . + n is equal to 
. (1)\r\n" );
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document.write( "For the proof, see the lessons\r\n" );
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document.write( " - Arithmetic progressions\r\n" );
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document.write( " - The proofs of the formulas for arithmetic progressions \r\n" );
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document.write( "in this site.\r\n" );
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document.write( "(a) Therefore, 1 + 2 + 3 + . . . + 24 = 
= 300.\r\n" );
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document.write( "(b) From the formula (1),\r\n" );
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document.write( " 
+ 
+ . . . + 
= 
= 
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document.write( "(c) Group the sum in this way\r\n" );
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document.write( " Sum = (1/1) + (1/2 + 2/2) + (1/3 + 2/3 + 3/3) + (1/4 + 2/4 + 3/4 + 4/4) + . . . (2)\r\n" );
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document.write( " We have 300 terms/addends in all and the number of terms in k-th separate parentheses is k.\r\n" );
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document.write( " Referring to the previous part (b) of this problem, we conclude that there are 24 groups in parentheses in the sum (2).\r\n" );
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document.write( " Each particular group (k-th group) has the sum equal to 
, according to part (b) of the solution.\r\n" );
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document.write( " In other words,\r\n" );
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document.write( " Sum = 
= 
= 
= 
= 
= 
= 324 
= 324.5. ANSWER\r\n" );
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document.write( "Solved.\r
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document.write( "Post-solution note\r
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document.write( " In this problem, its separate parts (a), (b) and (c) are logically inter-connected.\r
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document.write( "![\"\"](\"/images/stars/stars-13.gif\")
