Question 1209805
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Let a_1 + a_2 + a_3 + dots be an infinite geometric series with positive terms. 
If a_2 = 10, then find the smallest possible value of a_1 + a_2 + a_3.
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        This problem is simple and elementary,  and I will show below 

        a simple solution without using Calculus and/or derivatives.



<pre>
The fact that this geometric progression has positive terms tells us
that the first term {{{a[1]}}} is positive and the common ratio is positive, too.


So, the sum  {{{a[1] + a[2] + a[3]}}} can be presented in the form

    {{{a[2]/r}}} + {{{a[2]}}} + {{{a[2]*r}}} = {{{10/r}}} + 10 + 10*r.    (1)


We can identically transform this expression in the right side of (1) this way

    {{{10/r}}} + 10 + 10r = ({{{10/r}}} - 20 + 10r) + 30 = {{{(sqrt(10/r) - sqrt(10r))^2}}} + 30.    (2)


Now,  the part  {{{(sqrt(10/r) - sqrt(10r))^2}}}  is always greater than or equal to zero, 
since it is the square of real number.



Hence, this expression is minimal if and only if  

    {{{sqrt(10/r)}}} = {{{sqrt(10r)}}},    (3)

when  {{{(sqrt(10/r) - sqrt(10r))^2}}}  is equal to zero.



Square both sides in (3)

    {{{10/r}}} = 10r,

or

    {{{1/r}}}  = r,  -->  1 = r^2  -->  r = {{{sqrt(1)}}} = 1.



Hence, the sum (1) is minimal if and only if  r = 1.

Then the sum (1)  is   {{{10/1}}} + 10 + 10*1 = 10 + 10 + 10 = 30.



At this point, the solution is complete.


<U>ANSWER</U>.  The sum  {{{a[1] + a[2] + a[3]}}}  of geometric progression with positive terms 

         is  minimal if and only if  the common ratio r is 1.

         It is the case when all three terms of the progression are equal.

         For our case, this minimal value of the sum of the first three terms is 30, i.e. thrice its central term.
</pre>

Solved completely.


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As this problem is worded and presented, it considers only three first terms of the geometric progression.


Therefore, in the problem's formulation, there is no any need to consider an infinite progression.


Good style tells us to consider only three-term geometric progression from the very beginning.


Moreover, an infinite geometric progression with r= 1 diverges and its sum does not exist (is infinity).