document.write( "Question 868013: 1. find the sum of the first 100 000 odd numbers.
\n" ); document.write( "2. find the greatest possible product of two natural numbers whose sum is 41. \n" ); document.write( "

Algebra.Com's Answer #523532 by KMST(5328) $\"\"$ $\"About$

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The first 100,000 odd numbers are the odd numbers from
\n" ); document.write( " $\"2%2A1-1=2-1=1\"$ to $\"2%2A100\"$ $\"%22%2C000%22-1=200\"$ $\"%22%2C000%22-1=199\"$ $\"%22%2C999%22\"$ :
\n" ); document.write( "1, 3, 5, .... 199,993, 199,995, 199,997, 199,999.
\n" ); document.write( "The sum of such an arithmetic sequence
\n" ); document.write( "(or arithmetic progression, if that's what it's called in your class)
\n" ); document.write( "is 100,000 =100000= $\"10%5E5\"$ (the number of terms in that sum)
\n" ); document.write( "times half the sum of the first and last terms,
\n" ); document.write( " $\"%281%2B199999%29%2F2=200000%2F2=100000=10%5E5\"$ = 100,000.
\n" ); document.write( "So that sum is $\"100000%2A100000\"$ = 10,000,000,000,
\n" ); document.write( "or $\"10%5E5%2A10%5E5=10%5E10\"$ .
\n" ); document.write( "Your teacher may want to see formulas.
\n" ); document.write( "I do not know what formulas you were taught,
\n" ); document.write( "and I believe in understanding concepts rather than memorizing and blindly applying formulas,
\n" ); document.write( "but I can throw a few formulas for you to use as needed.
\n" ); document.write( "The same concepts can be expressed in many different ways.
\n" ); document.write( "An arithmetic sequence has a first term $\"a%5B1%5D\"$ ,
\n" ); document.write( "and each term is the term before plus a common difference $\"d\"$ ,
\n" ); document.write( "which can be written as
\n" ); document.write( " $\"a%5Bn%5D=a%5Bn-1%5D%2Bd\"$ .
\n" ); document.write( "As a consequence
\n" ); document.write( " $\"a%5B2%5D=a%5B1%5D%2Bd\"$
\n" ); document.write( " $\"a%5B3%5D=a%5B1%5D%2B2d\"$
\n" ); document.write( " $\"a%5Bn%5D=a%5B1%5D%2B%28n-1%29d\"$ ,
\n" ); document.write( "and of course, $\"a%5Bn-1%5D=a%5B1%5D%2B%28n-2%29d\"$
\n" ); document.write( "The sum of the first $\"n\"$ terms is
\n" ); document.write( " $\"S%5Bn%5D=a%5B1%5D%2Ba%5B2%5D%2B%22...%22%2Ba%5Bn-1%5D%2Ba%5Bn%5D\"$ =

\n" ); document.write( "That last expression contains a sum of $\"n\"$ 2-term sums, and all the 2-term sums are equal:
\n" ); document.write( "

,
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.
\n" ); document.write( "So $\"S%5Bn%5D=%281%2F2%29%28a%5B1%5D%2Ba%5Bn%5D%29%2An=n%2A%28%28a%5B1%5D%2Ba%5Bn%5D%29%2F2%29\"$ .
\n" ); document.write( "That is the number of terms times half the sum of the first and last terms,
\n" ); document.write( "or (in other words) $\"n\"$ times the average of the first and last terms.
\n" ); document.write( "An alternate formula is
\n" ); document.write( " $\"S%5Bn%5D=%281%2F2%29%282a%5B1%5D%2B%28n-1%29d%29%2An\"$ .
\n" ); document.write( "
\n" ); document.write( "If $\"x\"$ is one of two natural numbers whose sum is $\"41\"$ ,
\n" ); document.write( "then the other number is $\"41-x\"$ ,
\n" ); document.write( "and their product $\"y\"$ is
\n" ); document.write( " $\"y=%2841-x%29x=41x-x%5E2\"$ .
\n" ); document.write( "Traditionally, we prefer to rearrange the terms in the order $\"y=-x%5E2%2B41x\"$ .
\n" ); document.write( "We notice that $\"y\"$ is a quadratic function of $\"x\"$ ,
\n" ); document.write( "and its graph would look like this,
\n" ); document.write( " $\"graph%28300%2C300%2C-5%2C45%2C-500%2C500%2C-x%5E2%2B41x%29\"$ , with a maximum.
\n" ); document.write( "To find the maximum we can \"complete the square\":
\n" ); document.write( " $\"y=-x%5E2%2B41x\"$ <---> $\"y=-%28x%5E2-41x%2B%2841%2F2%29%5E2%29%2B%2841%2F2%29%5E2\"$ <---> $\"y=-%28x-41%2F2%29%5E2%2B%2841%2F2%29%5E2\"$
\n" ); document.write( "Alternately, we can use the formula that says that the maximum for $\"y=ax%5E2%2Bbx%2Bc\"$ occurs at $\"x=-b%2F2a\"$ .
\n" ); document.write( "If we could use any rational value for $\"x\"$ ,
\n" ); document.write( "the maximum would occur for $\"x=41%2F2=20.5\"$ ,
\n" ); document.write( "where we would have $\"y=%2841%2F2%29%5E2=41%5E2%2F2%5E2=1681%2F4=420.25\"$ .
\n" ); document.write( "However, since $\"x\"$ must be a natural number,
\n" ); document.write( "the maximum value for the product will happen for the natural numbers $\"x\"$ closest to $\"20.5\"$ :
\n" ); document.write( " $\"x=20\"$ ---> $\"41-x=41-20=21\"$
\n" ); document.write( "and $\"x=21\"$ ---> $\"41-x=41-21=20\"$ .
\n" ); document.write( "IN either case, the product will be
\n" ); document.write( " $\"20%2A21=highlight%28420%29\"$ \r
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