Prove, 1/n ? n - 1

Question

Prove or disprove. For all n in rational, 1/n ≠ n - 1

Answer 1

1/n ? n - 1

 

Multiply all terms by n

 

 1 ? n² - n

 

 0 ? n² - n - 1

 

Solve the quadratic

 

  n = ( - (-1) ± √ (1 - 4(1)(-1))) / 2(1)

 

  n = ( 1 ±  √ 5 ) / 2

 

√ 5 is an irrational number, therefore

 

(1/n) ≠ n - 1 for rational n's

Answer 2

Assume 1/n = n - 1 and solve this equation.

Note n cannot be zero.

Multiply throughout by n to get 1 = n²-n; solve this quadratic. If there is a Real answer (or two), you have it--otherwise no solution.