document.write( "Question 1083649: find the sum of the first n terms of the G.P 5+15+45+......... what is the smallest number of terms that will give a total of 1000. \n" ); document.write( "

Algebra.Com's Answer #697676 by KMST(5328) $\"\"$ $\"About$

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The first term is $\"b%5B1%5D=5\"$ .
\n" ); document.write( "The common ratio is r=15÷5=3 .
\n" ); document.write( "Term number $\"n\"$ is $\"b%5Bn%5D=5%2A3%5E%28n-1%29\"$ .
\n" ); document.write( "The sum of the first $\"n\"$ terms is
\n" ); document.write( " $\"b%5B1%5D%28r%5En-1%29%2F%28r-1%29=5%283%5En-1%29%2F%283-1%29=5.%283%5En-1%29%2F2\"$
\n" ); document.write( "If we want to find what is the smallest n
\n" ); document.write( "that will make the,sun of the first n terms no less than 100, our inequality is
\n" ); document.write( " $\"5%283%5En-1%29%2F2%3E=1000\"$
\n" ); document.write( " $\"3%5En-1%3E=1000%2A2%2F5\"$
\n" ); document.write( " $\"3%5En-1%3E=400\"$
\n" ); document.write( " $\"3%5En%3E=401\"$
\n" ); document.write( " $\"3%5E6=729\"$
\n" ); document.write( " $\"3%5E5=243\"$
\n" ); document.write( "The sum of first $\"n\"$ terms that is no less than 1000 is the sum of the first 6 terms:
\n" ); document.write( "5+15+45+135+405+1215.
\n" ); document.write( "The sum of some terms cannot be exactly 1000.
\n" ); document.write( "For the sum of some terms of the form $\"b%5Bk%2B1%5D=5%2A3%5Ek\"$
\n" ); document.write( "(with $\"k%3E=0\"$ being some integer)
\n" ); document.write( "to be exactly 1000,
\n" ); document.write( "the sum of all the $\"3%5Ek\"$ factors has to be 200.´
\n" ); document.write( "All those factors are multiples of 3, except $\"3%5E0=1\"$ .
\n" ); document.write( "and the sum of multiples of 3 cannot be either 200 or 199. \n" ); document.write( "

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