Question

Докажите, что при любом натуральном значении n значение выражения (n-1)(n+1)-(n-7)(n-5) кратно 12 У нас тема многочлены

Answer 1

(n-1)(n+1)-(n-7)(n-5)=n^2-1-n^2-35+12n=12n-36=12(n-3) что кратно 12

Answer 2

N2-1-(n2-5n-7n+35)=n2-1-n2+5n+7n-35=12n-36=12(н-2)
12:12 и 36:12 значит все делится на 12

Answer 3

Proof that (n-1)(n+1)-(n-7)(n-5) is a multiple of 12

To prove that the expression (n-1)(n+1)-(n-7)(n-5) is a multiple of 12 for any natural value of n, we can expand the expression and then factorize it to show that it is indeed divisible by 12.

Expanding the expression: ``` (n-1)(n+1) - (n-7)(n-5) = n^2 - n + n - 1 - (n^2 - 5n - 7n + 35) = n^2 - n + n - 1 - n^2 + 12n - 35 = 12n - 36 = 12(n - 3) ```

From the expanded expression, it's clear that (n-1)(n+1)-(n-7)(n-5) simplifies to 12 times (n-3), which means it is indeed a multiple of 12 for any natural value of n.

Conclusion: The expression (n-1)(n+1)-(n-7)(n-5) is always a multiple of 12 for any natural value of n.