by sweetshona

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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.

==========

NOW, the answer: ** 11th terms are **

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

That is that.