Question

Find the values of n and X in each of the following cases :

(i)  `sum _(i = 1)^n`(x_i - 12) = - 10 `sum _(i = 1)^n`(x_i - 3) = 62

(ii) `sum _(i = 1)^n` (x_i - 10) = 30  `sum _(i = 6)^n` (x_i - 6) = 150 .

Answer 1

(i)  Given `sum _(i = 1)^n`(x_n - 12) = - 10 

⇒` (x_1 - 12 ) + ( x _2 - 12 ) = ....... + (x_n - 12)  = - 10`

⇒ `(x_1 + X_2 + x_3+X_4 + x _5 + ...... + x_n) - ( 12 + 12 + 12 .........+12) = - 10`

⇒ `sumx - 12 _n = -10 ......... (1)`

And `sum _(i = 1 )^n(x_i - 3) = 62 `  `⇒  ( x_1 - 3) + ( x_2 - x_3 ) + ( x_3 - 3) + ....... + ( x_n - 3)` = 62.

⇒ ` ( x_1 + x_2 + .......... + x_n ) - (3 + 3 + 3 + 3 + ...... + 37)`= 62

⇒ `sum x - 3_n = 62`........ (2)

By subtracting equation (1) from equation (2)We get

` sumx - 3_n - sumx + 12_n = 62 + 10`

⇒ `9_n` = 72

⇒ `n = 72 / 9 = 8` 

Put value of n in equation (1)

`sumx - 12 xx 8 =-10`

⇒ `sumx - 96 = - 10`

⇒ `sumx =-10 + 96 = 86`

∴ x = `(sumx)/x  = 86/8 = 10 . 75`

(ii) Given `sum_(i - 1)^n (x_2 - 10) = 30`

⇒ ` ( x _1 -10) + ( x_2 -10) + ....... + ( x_n -10)` = 30

⇒ `(x _ 1 + x _ 2 + x _3 + ....... + x _ n) - ( 10 + 10 + 10 + ..... + 10 + ) = 30`

⇒ ` sumx -10_ n = 30       ........ (1)`

And `sum_ (i = 1)^n (x_i - 6) 150.`

⇒ `(x_1 - 6) + (x_2 - 6) + .... + (x_n - 6) = 150`

⇒ `( x_1 + x_2 + x_3 + .......... + x_n ) - (6 + 6 + 6 + ...... + 6) = 150`

⇒ `sumx - 6n = 150`      ....(2)

By subtracting equation (1) from equation (2)

`sum x - 6_n - sumx + 10_n = 150 - 30` 

⇒ `sumx - sumx + 4n = 120`

⇒ `n = 120/4`

⇒ n = 30

Put value of n in equation (1)

`sumx - 10 xx 30 = 30`

⇒`sumx -300 = 30`

⇒ `sumx = 30+300 = 330`

         ∴ x =`(sumx)/ n = 330 /30 = 11`.