document.write( "Question 1062832: Find the arithmetic and geometric mean of the series 1,2,4,8,16,....,2^n find also harmonic mean \n" ); document.write( "

Algebra.Com's Answer #677839 by ikleyn(52790) $\"\"$ $\"About$

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\n" ); document.write( "Find the arithmetic and geometric mean of the series 1,2,4,8,16,....,2^n find also harmonic mean
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document.write( "1.  To find the ARITHMETIC MEAN, find the sum of the first (n+1) terms of the geometric progression, and then divide the sum \r\n" );
document.write( "    by the number of terms (n+1).    You should divide by (n+1)  (and not by \"n\", because there are (n+1) terms in the sequence from 1 to .\r\n" );
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document.write( "    It is SO CLEAR that I will do not do it for (and instead of) you.\r\n" );
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document.write( "    It will be MUCH BETTER if you do it on your own.\r\n" );
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document.write( "2.  To find the GEOMETRIC MEAN, find the product of the first \"n\" terms of the progression, and then take the (n+1)-th root of the product. \r\n" );
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document.write( "    We take the (n+1)-th root of the product (and not the n-th degree) because there are n+1 multipliers (factors) in the product \r\n" );
document.write( "    from  to .\r\n" );
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document.write( "    The product of the first \"n\" terms is\r\n" );
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document.write( "    P =  = 2 in degree .\r\n" );
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document.write( "    This index  (I mean the upper index, which is \"the degree\") arises because \r\n" );
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document.write( "    the indexes of \"2\" are 0, 1 , 2, 3, . . . k, . . . n, and their sum is \r\n" );
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document.write( "    0 + 1 + 2 + 3 + . . . + n =  = the sum of the first \"n\" natural numbers.\r\n" );
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document.write( "    Again, P = 2 in degree .\r\n" );
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document.write( "    Then the (n+1)-th degree root of P is 2^(n/2), or, which is the same, .\r\n" );
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document.write( "    Thus the geometric mean is 2^(n/2) = .\r\n" );
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document.write( "3.  The HARMONIC MEAN is \r\n" );
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document.write( "         (H)\r\n" );
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document.write( "    where the denominator is the sum of inverse values to the original sequence.\r\n" );
document.write( "    The numerator is (n+1) because there are n+1 term in our original sequence 1, 2, 4, . . . .\r\n" );
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document.write( "    See the definition of the \"harmonic mean\" in this Wikipedia article.\r\n" );
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document.write( "    The denominator of the formula (H) above is the sum of the geometric sequence, again.\r\n" );
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document.write( "    So, it is very similar to what you just saw in the n.1 (#1) above.\r\n" );
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document.write( "    Follow to instructions of the #1.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "There is a bunch of lessons on arithmetic progressions in this site:\r
\n" ); document.write( "\n" ); document.write( "    - Arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - The proofs of the formulas for arithmetic progressions \r
\n" ); document.write( "\n" ); document.write( "    - Problems on arithmetic progressions \r
\n" ); document.write( "\n" ); document.write( "    - Word problems on arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction and arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - One characteristic property of arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on arithmetic progressions \r
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\n" ); document.write( "\n" ); document.write( "There is a bunch of lessons on geometric progressions, too:\r
\n" ); document.write( "\n" ); document.write( "    - Geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - The proofs of the formulas for geometric progressions \r
\n" ); document.write( "\n" ); document.write( "    - Problems on geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Word problems on geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - One characteristic property of geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Fresh, sweet and crispy problem on arithmetic and geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction and geometric progressions\r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction for sequences other than arithmetic or geometric\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topics
\n" ); document.write( "\"Arithmetic progressions\" and \"Geometric progressions\".\r
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