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
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.
2 answers
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
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
1 answer