1. **State the problem:** We are given a function $$f(x) = \frac{1}{50} \left[ c(x-1) + \frac{1}{2} \right]$$ for $$x=1,2,...,10$$ and the condition $$\sum_{x=1}^{10} f(x) = 1$$. We need to find the value of the constant $$c$$ and express $$f(x)$$ explicitly. 2. **Use the summation condition:** Substitute $$f(x)$$ into the summation: $$\sum_{x=1}^{10} \frac{1}{50} \left[ c(x-1) + \frac{1}{2} \right] = 1$$ 3. **Factor out constants:** $$\frac{1}{50} \left[ c \sum_{x=1}^{10} (x-1) + \sum_{x=1}^{10} \frac{1}{2} \right] = 1$$ 4. **Calculate the sums:** - $$\sum_{x=1}^{10} (x-1) = 0 + 1 + 2 + ... + 9 = 45$$ (sum of first 9 natural numbers) - $$\sum_{x=1}^{10} \frac{1}{2} = 10 \times \frac{1}{2} = 5$$ 5. **Substitute sums back:** $$\frac{1}{50} [45c + 5] = 1$$ 6. **Solve for $$c$$:** Multiply both sides by 50: $$45c + 5 = 50$$ Subtract 5: $$45c = 45$$ Divide by 45: $$c = 1$$ 7. **Write the explicit function:** Substitute $$c=1$$ back into $$f(x)$$: $$f(x) = \frac{1}{50} \left( (x-1) + \frac{1}{2} \right) = \frac{1}{50} \left( x - \frac{1}{2} \right)$$ **Final answer:** $$f(x) = \frac{1}{50} \left( x - \frac{1}{2} \right) \quad \text{for} \quad x=1,2,...,10$$