Question 1116634
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<pre>
S         = 2/5 + 6/25 + 10/125 + 14/625 + . . . 

S/5       =       2/25 +  6/125 + 10/625 + . . .     Take the difference

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S - (S/5) = 2/5 + 4/25 + 4/125  + 4/625 + . . . =


          = 2/5 + 4*(1/25 + 1/125 + 1/625 + . . . )


          = {{{2/5}}} + {{{4*(((1/25))/(1-1/5))}}} = {{{2/5}}} + {{{(4*5)/(25*4)}}} = {{{2/5 + 1/5}}} = {{{3/5}}}  ====>


{{{(4/5)*S}}} = {{{3/5}}}  ====>  S = {{{((3/5))/((4/5))}}} = {{{3/4}}} = 0.75.


<U>Answer</U>.  2/5 + 6/25 + 10/125 + 14/625 + . . .  = {{{3/4}}} = 0.75.


<U>Check</U>.   2/5 + 6/25 + 10/125 + 14/625          =      0.7424.
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Solved.