Question

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.

Answer 1

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

Answer 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