Question 1062832
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Find the arithmetic and geometric mean of the series 1,2,4,8,16,....,2^n find also harmonic mean
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<pre>
1.  To find the <U>ARITHMETIC MEAN</U>, find the sum of the first (n+1) terms of the geometric progression, and then divide the sum 
    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 {{{2^n}}}.

    It is SO CLEAR that I will do not do it for (and instead of) you.

    It will be MUCH BETTER if you do it on your own.
</pre>

<pre>
2.  To find the <U>GEOMETRIC MEAN</U>, find the product of the first "n" terms of the progression, and then take the (n+1)-th root of the product. 

    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 
    from {{{2^0}}} to {{{2^n}}}.

    The product of the first "n" terms is

    P = {{{1 * 2 * 4 * ellipsis * 2^k * ellipsis * 2^n}}} = 2 in degree {{{n*(n+1)/2}}}.

    This index {{{(n*(n+1))/2}}} (I mean the upper index, which is "the degree") arises because 

    the indexes of "2" are 0, 1 , 2, 3, . . . k, . . . n, and their sum is 

    0 + 1 + 2 + 3 + . . . + n = {{{(n*(n+1))/2}}} = the sum of the first "n" natural numbers.

    Again, P = 2 in degree {{{n*(n+1)/2}}}.

    Then the (n+1)-th degree root of P is 2^(n/2), or, which is the same, {{{sqrt(2^n)}}}.

    Thus the geometric mean is 2^(n/2) = {{{sqrt(2^n)}}}.
</pre>

<pre>
3.  The <U>HARMONIC MEAN</U> is 

    {{{(n+1) / (1 + 1/2 + 1/4 + ellipsis + 1/2^n)}}}     (H)

    where the denominator is the sum of inverse values to the original sequence.
    The numerator is (n+1) because there are n+1 term in our original sequence 1, 2, 4, . . . {{{2^n}}}.

    See the definition of the "harmonic mean" in <A HREF=https://en.wikipedia.org/wiki/Harmonic_mean>this</A> Wikipedia article.


    The denominator of the formula (H) above is the sum of the geometric sequence, again.

    So, it is very similar to what you just saw in the n.1 (#1) above.

    Follow to instructions of the #1.
</pre>

Solved.


There is a bunch of lessons on arithmetic progressions in this site:

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Arithmetic-progressions.lesson>Arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/The-proofs-of-the-formulas-for-arithmetic-progressions.lesson>The proofs of the formulas for arithmetic progressions</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Problems-on-arithmetic-progressions.lesson>Problems on arithmetic progressions</A>  

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Word-problems-on-arithmetic-progressions.lesson>Word problems on arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-and-arithmetic-progressions.lesson>Mathematical induction and arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/One-characteristic-property-of-arithmetic-progressions.lesson>One characteristic property of arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Solved-problems-on-arithmetic-progressions.lesson>Solved problems on arithmetic progressions</A> 


There is a bunch of lessons on geometric progressions, too:

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Geometric-progressions.lesson>Geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/The-proofs-of-the-formulas-for-geometric-progressions.lesson>The proofs of the formulas for geometric progressions</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Problems-on-geometric-progressions.lesson>Problems on geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Word-problems-on-geometric-progressions.lesson>Word problems on geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/One-characteristic-property-of-geometric-progressions.lesson>One characteristic property of geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Solved-problems-on-geometric-progressions.lesson>Solved problems on geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Fresh-sweet-and-crispy-problem-on-arithmetic-and-geometric-progressions.lesson>Fresh, sweet and crispy problem on arithmetic and geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-and-geometric-progressions.lesson>Mathematical induction and geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-for-sequences-other-than-arithmetic-or-geometric.lesson>Mathematical induction for sequences other than arithmetic or geometric</A>



Also, you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topics 
<U>"Arithmetic progressions"</U> and <U>"Geometric progressions"</U>.