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```        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.```
8 years ago

147 Points
```										Dear Tapasranjan
let first series is
S1 =a +ar1  +ar12  +ar13 +ar14  ...............................
and second series is
S1 =b +br2  +br22  +br23 +br24  ...............................
given   ar1 = br2
S1 =1 =a/1-r1
or a=1-r1
third term  ar12  =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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```
8 years ago
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