1. The problem is to simplify the summation \(\sum_{k=1}^n (5k + 2)\) correctly. 2. The formula for the sum of the first \(n\) natural numbers is \(\sum_{k=1}^n k = \frac{n(n+1)}{2}\). 3. Using linearity of summation, split the sum: \[\sum_{k=1}^n (5k + 2) = 5 \sum_{k=1}^n k + \sum_{k=1}^n 2\] 4. Substitute the formula for \(\sum_{k=1}^n k\) and simplify the constant sum: \[= 5 \cdot \frac{n(n+1)}{2} + 2n\] 5. Your mistake is in the last step where you wrote \(+ n\) instead of \(+ 2n\) because \(\sum_{k=1}^n 2 = 2 + 2 + \cdots + 2 = 2n\). 6. So the correct simplified expression is: \[\sum_{k=1}^n (5k + 2) = \frac{5n(n+1)}{2} + 2n\] 7. This is the final answer and the correction to your mistake.