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2015-01-10T04:23:51+05:30

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Terms of Series S =  a , a + d , a + 2d , a+3d, ....
Sums of series S  =  a,  2a + d,  (2a + 2d)3/2, (2a+3d)*4/2 ....
 
Series R = b, b +e , b + 2e, b + 3e, ...
Sums of series R are  =   b,  2b +e ,  (2b + 2e)*3/2, (2b+  3e)*4/2

We have four variables,  a, d, b , and e.  We need at least 3 equations to  establish relations among them.

Sn / Rn = (7 n + 1 ) / ( 4 n + 27)

Let n = 1
S1/R1  = a/b = 8/31
       =>   31 a - 8 b = 0     --- equation 1


S2 / R2  = (2 a + d) / (2 b + e)  = 15 / 35  = 3/7
       =>   14 a + 7 d - 6 b - 3 e = 0  --  equation 2

S3 / R3 =  (3a+3d)/(3b+3e) = 22 / 39
         =>  39 a + 39 d - 22 b - 22 e = 0  -- equation 3

S4 / R4 =  (2a+3d)/(2b+3e) = 29/43
           =>  86 a - 58 b + 129 d - 87 e = 0    --  equation 4

Solving these equations we get :

  4 * eq 3 – eq 4  gives  us :

         =>     e = 70 a – 30 b + 27 d    --- equation 5

 Substituting this in equation 2 gives,

        =>  42 b = 98 a + 37 d    -- equation 6

Substituting value of b from equation 1 , we get ,

           =>  a = 4/7 * d

   Hence,     b = 31/14 * d

            e = 4/7 * d

Hence the two series are :

Terms of series   S:      4d/7,  11d/7 , 18d/7, 25d/7, ...

     Sums are :  4d/7,  15d/7,  33d/7, 58d/7 ...

Terms of series   R:      31d/14,  39d/14,  47d/14,  55d/14, ...

   Sums are  :    31d/14,  70d/14, 117d/14,  172d/14, ...

The ratios will be matching . please check. 

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NOW,  the answer:   11th terms are

    Ratio = [ 74 d/7 ] / [ 111d /14 ]  =  148 / 111

That is that.

2 5 2
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