[50] For some positive integer k, the sum of the absolute values of all the integers from –k through k is 12. What is the value of k?

The problem is asking for the sum of the absolute values of all integers from -k to k.

Step 1: Understand the problem The sum of the absolute values of all integers from -k to k is the same as the sum of all integers from 0 to k, multiplied by 2. This is because the absolute value of a negative number is its positive counterpart.

Step 2: Set up the equation The sum of all integers from 1 to n is given by the formula n*(n+1)/2. So, the sum of all integers from 0 to k is k*(k+1)/2. Since we are considering the sum of the absolute values from -k to k, we multiply this sum by 2.

So, the equation is 2 * (k*(k+1)/2) = 12.

Step 3: Solve the equation Solving the equation gives k*(k+1) = 12. This simplifies to k^2 + k - 12 = 0.

Step 4: Factor the equation The equation factors to (k-3)(k+4) = 0.

Step 5: Solve for k Setting each factor equal to zero gives the solutions k = 3 and k = -4.

Since k is a positive integer, the only valid solution is k = 3.