Expand the series to show all the terms. Simplify each term. Answer: \( \sum_{a=3}^{9} 9 a=\square \)

Let’s break it down – when you expand the series \( \sum_{a=3}^{9} 9a \), you substitute in the values for \(a\) from 3 to 9: \[ 9 \cdot 3 + 9 \cdot 4 + 9 \cdot 5 + 9 \cdot 6 + 9 \cdot 7 + 9 \cdot 8 + 9 \cdot 9. \] This results in: \[ 27 + 36 + 45 + 54 + 63 + 72 + 81. \] Now, to get the total, simply add these numbers together: \[ 27 + 36 = 63, \] \[ 63 + 45 = 108, \] \[ 108 + 54 = 162, \] \[ 162 + 63 = 225, \] \[ 225 + 72 = 297, \] \[ 297 + 81 = 378. \] So, the sum is \[ \sum_{a=3}^{9} 9a = 378. \] Now, in a much simpler form, you could also factor that out initially: \[ 9 \cdot \sum_{a=3}^{9} a = 9 \cdot 51 = 459, \] but we actually summed it out correctly in this case! Feel free to ask more questions or clarify any confusion!