document.write( "Question 1089104: Find the sum of n term of the series.
\n" ); document.write( "1+4+13+40+........ . \n" ); document.write( "
Algebra.Com's Answer #703562 by KMST(5328)\"\" \"About 
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If you calculate the differences between consecutive terms,
\n" ); document.write( "you find that those differences form a geometric sequence
\n" ); document.write( "(or geometric progression, or GP, if those are the preferred names where you live).
\n" ); document.write( "The terms are
\n" ); document.write( "\"a%5B1%5D=1=3%5E0\" ,
\n" ); document.write( "\"a%5B2%5D=a%5B1%5D%2B3%5E1=1%2B3=4\" ,
\n" ); document.write( "\"a%5B3%5D=a%5B2%5D%2B3%5E2=4%2B9=13\" ,
\n" ); document.write( "\"a%5B4%5D=a%5B3%5D%2B3%5E3=13%2B27=40\" .
\n" ); document.write( "So, \"a%5Bn%5D=1%2B3%2B3%5E2%2B3%5E3%2B%22...%22%2B3%5E%28n-1%29\"
\n" ); document.write( "The differences form a geometric sequence/progression with common ratio r=3.
\n" ); document.write( "So, you can calculate each term as the sum of that geometric sequence/progression.
\n" ); document.write( "You can also figure it out from polynomial products,
\n" ); document.write( "knowing that \"%28x-1%29%2A%281%2Bx%2Bx%5E2%2Bx%5E3%2B%22...%22%2Bx%5E%28n-1%29%29=x%5En-1\" ,
\n" ); document.write( "and for \"x=3\" you have
\n" ); document.write( "\"%283-1%29%2A%281%2B3%2B3%5E2%2B3%5E3%2B%22...%22%2B3%5E%28n-1%29%29=3%5En-1\" .
\n" ); document.write( "Either way, you end up with
\n" ); document.write( "\"a%5Bn%5D=sum%283%5Ek%2Ck=0%2Cn-1%29\"\"%22=%22\"\"%283%5En-1%29%2F%283-1%29\"\"%22=%22\"\"%283%5En-1%29%2F2\"
\n" ); document.write( "Each term is the sum of all the terms before,
\n" ); document.write( "The sum of you have to calculate is
\n" ); document.write( "\"1%2B4%2B13%2B40%2B%22...%22%2Ba%5Bn%5D=sum%28a%5Bk%5D%2Ck=1%2Cn%29\"\"%22=%22\"
\n" ); document.write( " .
\n" ); document.write( "The sum in the formula is related to the one we had calculated before:
\n" ); document.write( " , so
\n" ); document.write( "\"1%2B4%2B13%2B40%2B%22...%22%2Ba%5Bn%5D=%281%2F2%29%283%283%5En-1%29%2F2%29-n%2F2\"\"%22=%22\"\"3%283%5En-1%29%2F4-n%2F2\" \n" ); document.write( "
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