How to simplify 57:19

How to simplify 57:19

57 to 19
=（57÷19）:（19÷19）
=3:1

How to do it? If x square minus 3x plus 1 equals 0, what is the value of x square divided by the fourth power of x plus the square of x plus 1?
If x square minus 3x plus 1 equals 0, then what is the value of x square divided by the fourth power of x plus the square of x plus 1?

Because x ^ 2-3x + 1 = 0, x ^ 2 = 3x-1 x ^ 4 + x ^ 2 + 1 = (3x-1) ^ 2 + x ^ 2 + 1 = 9x ^ 2-6x + 1 + x ^ 2 + 1 = 10x ^ 2 - (6x-2) = 10x ^ 2-2 (3x-1) = 10x ^ 2-2x ^ 2 = 8x ^ 2, so the result is x ^ 2 / (8x ^ 2) = 1 / 8

1×2+2×3+3×4+4×5+5×6+…… +19×20

1*2+2*3+3*4+4*5+5*6+6*7+7*8+…… +19*20=1*2+（2*3+3*4）+（4*5+5*6）+（6*7+7*8）+…… +（18*19+19*20） =2*1²+2*3²+2*5²+2*7²+2*9²+…… +2*19² =2*（1^2+3^2+5^2…… +19 ^ 2) and 1

Find [(1 ^ 4 + 1 / 4) (3 ^ 4 + 1 / 4) (5 ^ 4 + 1 / 4) (19^4+1/4)]/[(2^4+1/4)(4^4+1/4)(6^4+1/4)…… (20^4+1/4)]
Find [(1 ^ 4 + 1 / 4) (3 ^ 4 + 1 / 4) (5 ^ 4 + 1 / 4) (19^4+1/4)]/[(2^4+1/4)(4^4+1/4)(6^4+1/4)(20^4+1/4)]

A ^ 4 + 1 / 4 = (A & sup2; + 1 / 2) & sup2; - A & sup2; = (A & sup2; + A + 1 / 2) (A & sup2; - A + 1 / 2) = [a (a + 1) + 1 / 2] [a (A-1) + 1 / 2] so [(1 ^ 4 + 1 / 4) (3 ^ 4 + 1 / 4) (5 ^ 4 + 1 / 4) (19^4+1/4)]/[(2^4+1/4)(4^4+1/4)(6^4+1/4)…… (20^4+1/4)]=[(1*2+1/2)*(1*0+1...

1/2*3+1/3*4+1/4*5+1/5*6.+1/19*20

1/2*3+1/3*4+1/4*5+1/5*6.+1/19*20
=1/2-1/3+1/3-1/4+...+1/19-1/20
=1/2-1/20
=9/20

(（1^4+1/4)(3^4+1/4)(5^4+1/4).(19^4+1/4))/((2^4+1/4)(4^4+1/4)(6^4+1/4).(20^4+1/4))

A ^ 4 + 1 / 4 = (A & sup2; + 1 / 2) & sup2; - A & sup2; = (A & sup2; + A + 1 / 2) (A & sup2; - A + 1 / 2) = [a (a + 1) + 1 / 2] [a (A-1) + 1 / 2] so [(1 ^ 4 + 1 / 4) (3 ^ 4 + 1 / 4) (5 ^ 4 + 1 / 4) (19^4+1/4)]/[(2^4+1/4)(4^4+1/4)(6^4+1/4)…… (20^4+1/4)]=[(1*2+1/2)*(1*0+1...

1+2+3.+18+19+20=

210

Simple calculation: 1 / (2 × 3) + 1 / (3 × 4) + 1 / (4 × 5) + +1/（18×19）+1/（19×20）

Do it by splitting
Become
1/2-1/3+1/3-1/4+...+1/19-1/20
In the end, all the middle items were crossed out, leaving only one
1/2-1/20=9/20

20-19+18-17+…… +4-3+2-1=?
20-19+18-17+…… +4-3 + 2-1 is equal to what.. help me to list the formula.. thank you

20-19+18-17+…… +4-3+2-1
=(20-19)+(18-17)+…… +(4-3)+(2-1)
=1+1+...+1
=10

1/(1*2*3*4)+1/(2*3*4*5)+1/(3*4*5*6)+.+1/(17+18+19+20)=?
1/(1*2*3*4)+1/(2*3*4*5)+1/(3*4*5*6)......+1/(17*18*19*20)=?

1/(1*2*3*4)=1/3(1/1*2*3-1/2*3*4)1/(2*3*4*5)=1/3(1/2*3*4-1/3*4*5)......1/(1*2*3*4)+1/(2*3*4*5)+1/(3*4*5*6).+1/(17*18*19*20)=1/3(1/1*2*3-1/18*19*20)=1/3*(3*19*20-1)/(18*19*20)=1139/20520