Let $x = 111 . . . . 11$ ( $20$ digits)
$y = 333 . . . . 33$ ( $10$ digits)
and $z = 222 . . . . 22$ ( $10$ digits),
The value of $\frac{x - y^{2}}{z}$ .

Open in App

Solution

The d-digit integer consisting only of $9^{'} s$ can be written $10^{d} - 1$ . set
$x = \frac{1}{9} (10^{2 d} - 1), y = \frac{1}{3} (10^{d} - 1), z = \frac{2}{9} (10^{d} - 1)$
we claim that $x - z = y^{2} .$ Indeed,
$x - z = \frac{1}{9} (10^{d} - 1) ((10^{d} + 1) - 2) = (\frac{1}{3} (10^{d} - 1))^{2} = y^{2} .$
Thus, for any $d \geq 1, (x - y^{2}) / z = 1.$ The given set of numbers corresponds to $d = 10. ■$

Suggest Corrections

Similar questions

Q. If

x = 111 \dots \dots 1 (20 digits),

y = 333 \dots \dots 3 (10 digits)

and

z = 222 \dots \dots 2 (10 digits)

, then

x - z

is equal to

Q. If

x = 111 \dots \dots 1 (20 digits),

y = 333 \dots \dots 3 (10 digits)

and

z = 222 \dots \dots 2 (10 digits)

, then

x - z

is equal to

In the reaction
$x B r O_{3}^{-} + y C r^{3 +} + z H_{2} O \to B r_{2} + H C r O_{4}^{-} + H^{+}$ .
The values of x, y & z respectively in the balanced equation are

If the 1st (Hundred's place), 2nd(Ten's place) and 3rd(Unit's place) digits of a 3-digit number are taken as x , y and z respectively,

how do we write the general form of the number when it is reversed?

Q. Let the positive number of solutions of

x + y + z + w = 20

under the following conditions:

(i)

zero values of

x, y, z, w

are included be "k"

(i i)

zero values are excluded be "m"
Find sum of digits of m+k ?

Join BYJU'S Learning Program

Let x=111....11 (20 digits) y=333....33 (10 digits)and z=222....22 (10 digits),The value of x−y2z.

The d-digit integer consisting only of 9′s can be written 10d−1. set x=19(102d−1),y=13(10d−1),z=29(10d−1)we claim that x−z=y2. Indeed,x−z=19(10d−1)((10d+1)−2)=(13(10d−1))2=y2.Thus, for any d≥1,(x−y2)/z=1. The given set of numbers corresponds to d=10.■

Let $x = 111 . . . . 11$ ( $20$ digits)
$y = 333 . . . . 33$ ( $10$ digits)
and $z = 222 . . . . 22$ ( $10$ digits),
The value of $\frac{x - y^{2}}{z}$ .