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Tapasranjan Das Grade: 12
        

Sir please post the solution of the following problem _


Q:-Two distinct,real,infinite geometric series each have a sum of 1 and have the same second term. Third term of one of the series is 1/8. If the second term of both series can be written  in the form ((√m) -n)/p , where m,n and p are positive integers and m is not divisible by the square of any prime,find the value of 100m+10n+p.

7 years ago

Answers : (1)

Badiuddin askIITians.ismu Expert
147 Points
										

Dear Tapasranjan


let first series is


S1 =a +ar1  +ar1+ar13 +ar14  ...............................


and second series is


S1 =b +br2  +br2+br23 +br24  ...............................


given   ar1 = br2


 S1 =1 =a/1-r1


  or a=1-r1


third term  ar1=1/8


put value of a


 (1-r1)r12 =1/8


or  r13 +r12 +1/8=0


by inspection one value of r1 = 1/2


devide above equation be r1 -1/2 =0


 remaining 2 roots are the roots of equation  4r12 -2r1-1=0


 r1= (1±√5)/4


second term  =ar1 =(1-r1)r1


                           =r1 - r12


                                  =r1-(r1 /2  +1/4)


                             =r1/2 -1/4


               put value of r1=(1+√5)/4


    second term  = (√5 -1)/8    other value of r will not give this form


 so m=5   , n=1  ,p=8


100m+10n+p = 518


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Badiuddin

7 years ago
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