Question

Докажите что значение выражения (n-1)(n 1)-(n-7)(n 3) кратно 4 при всех натуральных значениях n

Answer 1

= n²-2n+1-n²+7n+3n-21=8n-20=4(2n-5)
один из сомножителей делится на 4, значит и все произведение делится на 4.
Хотя в условии в четвертой скобке нет знака, поставил наобум

Answer 2

Checking the Divisibility of the Expression

To prove that the expression (n-1)(n+1)-(n-7)(n+3) is divisible by 4 for all natural values of n, we can expand the expression and simplify it to demonstrate its divisibility.

Expanding the expression: (n-1)(n+1) - (n-7)(n+3)

Expanding the terms: = n^2 - n + n - 1 - (n^2 + 3n - 7n - 21) = n^2 - n + n - 1 - (n^2 - 4n - 21) = n^2 - n + n - 1 - n^2 + 4n + 21 = -1 + 4n + 21 = 4n + 20

Proving Divisibility by 4

To prove that the expression 4n + 20 is divisible by 4 for all natural values of n, we can factor out 4 from the expression.

Factoring out 4: 4n + 20 = 4(n + 5)

As we can see, the expression 4n + 20 can be expressed as 4(n + 5), where n + 5 is an integer for all natural values of n. Therefore, the expression is divisible by 4 for all natural values of n.

This proves that the expression (n-1)(n+1)-(n-7)(n+3) is indeed divisible by 4 for all natural values of n.

[[SOURCE #4]]