Algebra.Com's Answer #851937 by ikleyn(52778)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
.
\n" );
document.write( "Let a_1 + a_2 + a_3 + dots be an infinite geometric series with positive terms.
\n" );
document.write( "If a_2 = 10, then find the smallest possible value of a_1 + a_2 + a_3.
\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " This problem is simple and elementary, and I will show below \r
\n" );
document.write( "\n" );
document.write( " a simple solution without using Calculus and/or derivatives.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "\r\n" );
document.write( "The fact that this geometric progression has positive terms tells us\r\n" );
document.write( "that the first term  is positive and the common ratio is positive, too.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, the sum  can be presented in the form\r\n" );
document.write( "\r\n" );
document.write( "  +  +  =  + 10 + 10*r. (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We can identically transform this expression in the right side of (1) this way\r\n" );
document.write( "\r\n" );
document.write( " + 10 + 10r = ( - 20 + 10r) + 30 =  + 30. (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now, the part is always greater than or equal to zero, \r\n" );
document.write( "since it is the square of real number.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Hence, this expression is minimal if and only if \r\n" );
document.write( "\r\n" );
document.write( "  =  , (3)\r\n" );
document.write( "\r\n" );
document.write( "when is equal to zero.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Square both sides in (3)\r\n" );
document.write( "\r\n" );
document.write( " = 10r,\r\n" );
document.write( "\r\n" );
document.write( "or\r\n" );
document.write( "\r\n" );
document.write( "  = r, --> 1 = r^2 --> r =  = 1.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Hence, the sum (1) is minimal if and only if r = 1.\r\n" );
document.write( "\r\n" );
document.write( "Then the sum (1) is  + 10 + 10*1 = 10 + 10 + 10 = 30.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "At this point, the solution is complete.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER. The sum of geometric progression with positive terms \r\n" );
document.write( "\r\n" );
document.write( " is minimal if and only if the common ratio r is 1.\r\n" );
document.write( "\r\n" );
document.write( " It is the case when all three terms of the progression are equal.\r\n" );
document.write( "\r\n" );
document.write( " For our case, this minimal value of the sum of the first three terms is 30, i.e. thrice its central term.\r\n" );
document.write( " \r
\n" );
document.write( "\n" );
document.write( "Solved completely.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "----------------------------\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "As this problem is worded and presented, it considers only three first terms of the geometric progression.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Therefore, in the problem's formulation, there is no any need to consider an infinite progression.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Good style tells us to consider only three-term geometric progression from the very beginning.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Moreover, an infinite geometric progression with r= 1 diverges and its sum does not exist (is infinity).\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " \n" );
document.write( " |