A partial sum of an arithmetic sequence is given. Find the sum. 16(5 + 0.28k)k = 0

The given equation is 16(5 + 0.28k)k = 0.

Step 1: Distribute the k into the parentheses to get 16(5k + 0.28k^2) = 0.

Step 2: Distribute the 16 into the parentheses to get 80k + 4.48k^2 = 0.

Step 3: This is a quadratic equation in the form of ax^2 + bx + c = 0. To solve for k, we can use the quadratic formula, which is k = [-b ± sqrt(b^2 - 4ac)] / (2a). In this case, a = 4.48, b = 80, and c = 0.

Step 4: Plug the values of a, b, and c into the quadratic formula to get k = [-80 ± sqrt((80)^2 - 44.480)] / (2*4.48).

Step 5: Simplify the equation to get k = [-80 ± sqrt(6400)] / 8.96.

Step 6: Solve for k to get k = -80/8.96 = -8.92857143 and k = 80/8.96 = 8.92857143.

So, the solutions for k are -8.92857143 and 8.92857143. However, since k cannot be negative in the context of a sum of a sequence, the only valid solution is k = 8.92857143.