Question

0.125n! = (n-1)! - 90

Answer 1

N!=1*2*3*....*(n-1)*n  где n - целое
0! = 1
методом подбора решать будем
Заметим что 1/8 * n! будет целым только при n>=4 тогда 0.125*2*4 = 1 
(n-1)! > 90 (факториал положителен) n>5 (5!>90)
4! = 1*2*3*4 = 24
5! = 24*5=120
6! = 120*6=720
7! = 720*7=5040
ну и проверим начиная с 6
0.125*6!=0.125*720 = 90   5!-90 = 120-90= 30 Нет
проверяем 7
0.125*7! =  0.125*5040 = 630   6! - 90 = 720 - 90 = 630 ДА
Итак n=7

Answer 2

To solve the equation 0.125n! = (n-1)! - 90, we can start by simplifying the factorials and then solving for 'n'. Let's go step by step:

Step 1: Express the factorials. n! = n × (n-1) × (n-2) × ... × 3 × 2 × 1 (n-1)! = (n-1) × (n-2) × ... × 3 × 2 × 1

Step 2: Substitute the factorials into the equation. 0.125n! = 0.125 × n × (n-1) × (n-2) × ... × 3 × 2 × 1 (n-1)! - 90 = (n-1) × (n-2) × ... × 3 × 2 × 1 - 90

Step 3: Simplify the equation. 0.125 × n! = (n-1)! - 90

Now, let's divide both sides by 0.125 to get rid of the coefficient:

n! = (n-1)! / 0.125 - 90

Next, we can simplify (n-1)! / 0.125:

n! = 8 * (n-1)! - 90

Now, we can equate the factorials:

n × (n-1) × (n-2) × ... × 3 × 2 × 1 = 8 * (n-1) × (n-2) × ... × 3 × 2 × 1 - 90

Notice that we can cancel out (n-1) × (n-2) × ... × 3 × 2 × 1 from both sides:

n = 8 - 90 n = -82

So, the solution for 'n' that satisfies the equation is -82. However, it's worth noting that factorials are generally defined only for non-negative integers, so this solution might not be meaningful in the context of factorials. It is possible that there was a mistake or misunderstanding in the original equation or that it was intended for a different mathematical domain.