The problem shown in the image asks for the sum of the series given by:

$\sum_{i=1}^{6} (8i + 9)$

Let's break this down step-by-step:

1. Series Expansion:

   • When $i = 1$: $8(1) + 9 = 8 + 9 = 17$
   • When $i = 2$: $8(2) + 9 = 16 + 9 = 25$
   • When $i = 3$: $8(3) + 9 = 24 + 9 = 33$
   • When $i = 4$: $8(4) + 9 = 32 + 9 = 41$
   • When $i = 5$: $8(5) + 9 = 40 + 9 = 49$
   • When $i = 6$: $8(6) + 9 = 48 + 9 = 57$

2. Sum the Values: $17 + 25 + 33 + 41 + 49 + 57 = 222$

So, the sum of the series is 222.

Would you like more details on the summation process or verification using another method?

Related Questions:

1. How would the sum change if the upper limit of the summation was increased?
2. What is the general formula for a summation of a linear function like this?
3. Can this summation be solved using any shortcut methods (e.g., arithmetic series)?
4. What is the significance of summation notation in calculus?
5. How can graphing utilities like calculators or software help in verifying summation results?

Tip: Always check for common patterns in summations, such as arithmetic sequences, to simplify calculations.