$\text{Exercise 6}$
`a) A = 22499...9100...09` $\text{(n - 2 digits 9, n digits 0)}$
`= 224. 10^(2n) + 99...9. 10^(n + 2) + 100...00 + 9` $\text{(n - 2 digits 9, n + 1 digits 0)}$
`= 225. 10^(2n) - 10^(2n) + (10^(n - 2) - 1). 10^(n + 2) + 10^(n + 1) + 9`
`= (15. 10^n)^2 - 10^(2n) + 10^2n - 10^(n + 2) + 10^(n + 1) + 9`
`= (15. 10^n)^2 - 10^n. 100 + 10^n. 10 + 9`
`= (15. 10^n)^2 - 90. 10^n + 9`
`= (15. 10^n)^2 - 2. 15. 10^n. 3 + 3^2`
`= (15. 10^n + 3)^2` $\text{is the square number}$
`<=>` $\text{What have to proof}$
`b) B = 11...155...56` $\text{(n digits 1, n - 1 digits 5)}$
`= 11...1. 10^n + 55...55 + 1` $\text{(n digits 1, n digits 5)}$
`= (10^n - 1)/9. 10^n + (5(10^n - 1))/9 + 1`
`= (10^(2n) - 10^n)/9 + (5. 10^n - 5)/9 + 9/9`
`= (10^(2n) - 10^n + 5. 10^n - 5 + 9)/9`
`= ((10^n)^2 + 4. 10^n + 4)/9`
`= ((10^n)^2 + 2. 10^n. 2 + 2^2)/9`
`= (10^n + 2)^2/3^2`
`= ((10^n + 2)/3)^2` $\text{is the square number}$
`<=>` $\text{What have to proof}$
