\( \sum_{t=b}^{40} 5 \) Evaluate

To evaluate the summation: \[ \sum_{t=b}^{40} 5 \] This represents adding the constant value 5 for each integer value of \( t \) starting from \( t = b \) up to \( t = 40 \). ### Step-by-Step Evaluation: 1. **Identify the Number of Terms:** The number of integer values from \( t = b \) to \( t = 40 \) is: \[ \text{Number of terms} = 40 - b + 1 = 41 - b \] (We add 1 because both endpoints \( b \) and \( 40 \) are included.) 2. **Calculate the Sum:** Since each term in the summation is 5, the total sum is: \[ \text{Sum} = 5 \times (\text{Number of terms}) = 5 \times (41 - b) \] 3. **Final Expression:** \[ \sum_{t=b}^{40} 5 = 5(41 - b) \] ### Example: If \( b = 30 \), then: \[ \sum_{t=30}^{40} 5 = 5(41 - 30) = 5 \times 11 = 55 \] ### Conclusion: The evaluated sum is: \[ \sum_{t=b}^{40} 5 = 5(41 - b) \]