SOLUTION: Find the arithmetic and geometric mean of the series 1,2,4,8,16,....,2^n find also harmonic mean

1.  To find the ARITHMETIC MEAN, 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 .

    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.

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. 

    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  to .

    The product of the first "n" terms is

    P =  = 2 in degree .

    This index  (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 =  = the sum of the first "n" natural numbers.

    Again, P = 2 in degree .

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

    Thus the geometric mean is 2^(n/2) = .

3.  The HARMONIC MEAN is 

         (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, . . . .

    See the definition of the "harmonic mean" in this 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.