Consider the pattern:
11 - 2 = 3^2
1111 - 22 = 33^2
111111 - 222 = 333^2
.....
x - y = 33,333,333^2
Find the values of x and y.
Write your problem as:
xn + yn = zn
11 , 1111 , 111111
x1 = 11
x1 = 99 / 9
x1 = ( 100 - 1 ) / 9
x1 = ( 100 ^ 1 - 1 ) / 9
x2 = 1111
x2 = 9999 / 9
x2 = ( 10000 - 1 ) / 9
x2 = ( 100 ^ 2 - 1 ) / 9
x3 = 111111
x2 = 999999 / 9
x2 = ( 1000000 - 1 ) / 9
x2 = ( 100 ^ 3 - 1 ) / 9
etc.
xn = ( 1 / 9 ) [ ( 100 ^ n ) - 1 ]
2 , 22 , 222...
y1 = 2 = ( 10 ^ 1 - 1 ) * 2 / 9 = ( 10 - 1 ) * 2 / 9 = 9 * 2 / 9
y2 = 22 = ( 10 ^ 2 - 1 ) * 2 / 9 = ( 100 - 1 ) * 2 / 9 = 99 * 2 / 9 = 11 * 2
y3 = 222 = ( 10 ^ 3 - 1 ) * 2 / 9 = ( 1000 - 1 ) * 2 / 9 = 999 * 2 / 9 = 111 * 2
etc.
yn = ( 2 / 9 ) (10 ^ n - 1)
3 ^ 2 , 33 ^ 2 , 333 ^ 2
z1 = 3 ^ 2 = ( 10 - 1 ) ^ 2 / 9 = 9 ^ 2 / 9 = 81 / 9 = 9 = 3 ^ 2
z2 = 33 ^ 2 = ( 100 - 1 ) ^ 2 / 9 = 99 ^ 2 / 9 = 9801 / 9 = 33 ^ 2
z3 = 333 ^ 2 = ( 1000 - 1 ) ^ 2 / 9 = 999 ^ 2 / 9 = 998001 / 9 = 333 ^ 2
etc.
zn = ( 1 / 9 ) ( 10 ^ n - 1) ^ 2
So:
xn + yn = zn
( 1 / 9 ) [ ( 100 ^ n ) - 1 ] + ( 2 / 9 ) (10 ^ n - 1) = ( 1 / 9 ) ( 10 ^ n - 1) ^ 2
zn = ( 1 / 9 ) ( 10 ^ n - 1) ^ 2 = 33,333,333^2 = 1,111,111,088,888,889
( 1 / 9 ) ( 10 ^ n - 1) ^ 2 = 1,111,111,088,888,889 Multiply both sides by 9
( 10 ^ n - 1) ^ 2 = 9,999,999,800,000,001 Take square root of both sides
10 ^ n - 1 = 99,999,999 Add 1 to both sides
10 ^ n - 1 + 1 = 99,999,999 + 1
10 ^ n = 10,0000,000 Take the logarithm base 10 of both sides
n = 8
Now:
n = 8
xn + yn = zn
x8 + y8 = z8
( 1 / 9 ) [ ( 100 ^ n ) - 1 ] + ( 2 / 9 ) (10 ^ n - 1) = 33,333,333 ^ 2
( 1 / 9 ) [ ( 100 ^ 8 ) - 1 ] + ( 2 / 9 ) (10 ^ 8 - 1) = 33,333,333 ^ 2
( 1 / 9 ) ( 10,000,000,000,000,000 - 1 ] + ( 2 / 9 ) ( 100,000,000 - 1) = 33,333,333 ^ 2
( 1 / 9 ) ( 9,999,999,999,999,999 ] + ( 2 / 9 ) ( 99 999,999 ) = 33,333,333 ^ 2
99,999,999 / 9 + 199,999,998 / 9 = 33,333,333 ^ 2
1,111,111,111,111,111 + 22,222,222 = 33,333,333 ^ 2
My big typo!!!
xn - yn = zn
Now:
n = 8
xn - yn = zn
x8 - y8 = z8
( 1 / 9 ) [ ( 100 ^ n ) - 1 ] - ( 2 / 9 ) (10 ^ n - 1) = 33,333,333 ^ 2
( 1 / 9 ) [ ( 100 ^ 8 ) - 1 ] - ( 2 / 9 ) (10 ^ 8 - 1) = 33,333,333 ^ 2
( 1 / 9 ) ( 10,000,000,000,000,000 - 1 ] - ( 2 / 9 ) ( 100,000,000 - 1) = 33,333,333 ^ 2
( 1 / 9 ) ( 9,999,999,999,999,999 ] - ( 2 / 9 ) ( 99 999,999 ) = 33,333,333 ^ 2
99,999,999 / 9 - 199,999,998 / 9 = 33,333,333 ^ 2
1,111,111,111,111,111 - 22,222,222 = 33,333,333 ^ 2
Let's observe the pattern in the given examples:
11 - 2 = 3^2
1111 - 22 = 33^2
111111 - 222 = 333^2
From these examples, we can notice that each time, the number of digits in the left-hand side (LHS) of the equation is increasing by 2. Similarly, the right-hand side (RHS) of the equation is increasing by 1.
Applying this pattern to our equation, we can determine the number of digits in the LHS as:
x = 1111111... (where the number of digits is increasing by 2)
To find the number of digits, we can subtract 33,333,333 from x to get the remaining digits:
x - 33,333,333 = 1111111... - 222
Now, let's solve for x.
x - 33,333,333 = 1111111... - 222
x = 1111111... + 33,333,333 - 222
Since we want to find the value of x and y, let's split the equation further:
x = 1111111... + 33,333,111 - 222
y = 33,333,111
Therefore, the values of x and y are:
x = 1111111... + 33,333,111 - 222
y = 33,333,111
To find the values of x and y, we need to understand the pattern in the given examples and then apply it to the equation x - y = 33,333,333^2.
In the given pattern, the number 11 is subtracted from 2 to form the perfect square of 3 (3^2 = 9). Similarly, 22 is subtracted from 1111 to form the perfect square of 33 (33^2 = 1,089) and 222 is subtracted from 111111 to form the perfect square of 333 (333^2 = 110,889).
We can observe that for each perfect square, the number being subtracted (y) is twice the number of digits that are repeated, and the repeating digits are all ones. In other words:
For 3^2: Number of digits repeated = 1, y = 11 * 2 = 22
For 33^2: Number of digits repeated = 2, y = 1111 * 2 = 2222
For 333^2: Number of digits repeated = 3, y = 111111 * 2 = 222222
By following this pattern, we can conclude that for the equation x - y = 33,333,333^2, the number of repeated digits would be 8 (since 33,333,333 has 8 digits). Thus, y would be the number obtained by repeating the digit 2 eight times, i.e., y = 222,222,222.
Now, to find the value of x, we can simply add y to 33,333,333 squared:
x = y + (33,333,333^2)
x = 222,222,222 + (1,111,111,110,111,111)
Therefore, the values of x and y in the equation x - y = 33,333,333^2 are:
x = 1,111,111,332,111,333
y = 222,222,222