Question

Buktikan bahwa
a. n > 1, maka
n! - ( n-1 )! = ( n-1)! (n-1)
b. n > 3, maka
n! - (n-3)! = (n-3)! (n pangkat 3 - 3n pangkat 2 = 2n-1 )

Answer 1

N! - (n - 1)!

= n(n - 1)! - (n - 1)!
dengan memisalkan (n - 1)! = a

na - a = a(n - 1)
= (n - 1)! (n - 1)
terbukti

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n! - (n - 3)!
= n(n - 1)(n - 2)(n - 3)! - (n - 3)!
dengan memisalkan (n - 3)! = a

n(n - 1)(n - 2)a - a = a(n(n - 1)(n - 2) - 1)
= (n - 3)! (n(n - 1)(n - 2) - 1)
= (n - 3)! ((n² - n)(n - 2) - 1)
= (n - 3)! (n³ - 2n² - n² + 2n - 1)
= (n - 3)! (n³ - 3n² + 2n - 1)
terbukti