CPhill

1/17  is a repeating decimal with a period of 16  =  0.0588235294117647......

The 4000th digit  is  7

1n^5 - 38 n^4 - 188 n^3 + 12006 n^2 - 11781 n + 0

https://web2.0calc.com/questions/right-triangle_44

a1 =  1

a2 = 3

a3 = 7

a4  = 17

a5 = 41

a6 = 99

a7  = 239

a8 = 577

577 mod 3  =  1

Obviously, the hundreds and the units digits must be the same      

Let each of these = x

Let the tens  digit = y

So

x + x + xy = y + 8

2x + xy  - y  = 8       take the product of the  coefficients on x,y  = 2(-1) = -2  and add to each side

2x + xy - y - 2 = 8 - 2        factor the left side  and simplify the right

(x - 1) ( y + 2)  =  6

x       y

2      4

3      1

4      0

The three-digit numbers are

242

313

404

244224 mod 64  =  0

https://web2.0calc.com/questions/help-help_52

PA = sqrt (5^2 + 7^2)  = sqrt  74 = AQ

PQ = sqrt (2^2 + 2^2)  = sqrt 8

Law of Cosines

PQ^2  = PA^2 + PQ^2  - 2(PA * PQ)  cos (PAQ)

8 = 74 + 74  - 2 ( 74)  cos (PAQ)

cos (PAQ)  =  [ 8 - 74 -74 ] / [ -2 * 74 ]   =  35 / 37

sin (PAQ)  = sqrt [ 37^2 - 35^2 ] /  37 =  sqrt [ 144 ] / 37 =   12 / 37

https://web2.0calc.com/questions/help-meee_22

a + b = 1    square both sides

a^2 + b^2 + 2ab = 1

     2  + 2ab = 1

2ab = -1

ab = -1/2

a^3 + b^3  =  (a + b) ( a^2 + b^2 -ab) = (1) (2 - -1/2) = 2 + 1/2  =  5/2