Question

Найдите все целые значения, которые может принимать дробь 12n+109/3n+8 при натуральных значениях n. В ответ запишите сумму найденных значений.ПОМОГИТЕ ПОЖАЛУЙСТА. Зарание спс

Answer 1

При всех значениях n надо, что в результате получалась сократимая дробь до целого числа. Такое значение мы получим, если n=1

Answer 2

Finding the Values of n for the Fraction 12n + 109 / 3n + 8

To find the values of n for which the fraction 12n + 109 / 3n + 8 takes integer values, we need to solve the equation:

12n + 109 = k(3n + 8)

where k is an integer.

Let's simplify the equation:

12n + 109 = 3kn + 8k

Rearranging the terms:

12n - 3kn = 8k - 109

Factoring out n:

n(12 - 3k) = 8k - 109

Now, we have two cases to consider:

Case 1: 12 - 3k ≠ 0

In this case, we can divide both sides of the equation by (12 - 3k):

n = (8k - 109) / (12 - 3k)

To find the values of n, we need to find the values of k that make the right side of the equation an integer.

Case 2: 12 - 3k = 0

In this case, we have 12 - 3k = 0, which implies 3k = 12, and k = 4.

Substituting k = 4 into the original equation, we get:

12n + 109 = 4(3n + 8)

12n + 109 = 12n + 32

109 = 32

This equation is not true, which means there are no solutions for n when 12 - 3k = 0.

Therefore, we only need to consider Case 1.

To find the values of n, we can substitute different values of k and check if the right side of the equation is an integer.

Let's try a few values of k:

- For k = 1, n = (8(1) - 109) / (12 - 3(1)) = (-101) / 9 = -11.22 (not an integer) - For k = 2, n = (8(2) - 109) / (12 - 3(2)) = (-93) / 6 = -15.5 (not an integer) - For k = 3, n = (8(3) - 109) / (12 - 3(3)) = (-77) / 3 = -25.67 (not an integer) - For k = 4, n = (8(4) - 109) / (12 - 3(4)) = (-69) / 0 (undefined)

Based on these calculations, we can see that there are no integer values of n that satisfy the equation 12n + 109 / 3n + 8.

Therefore, the sum of the values of n is 0.

I hope this helps! Let me know if you have any further questions.