Evaluation of the Sum from 1 to 9 of the Expression 5k + 8

Evaluation of the Sum

The objective is to evaluate the sum given by the notation:

ext{Evaluate the sum }
um{9} imes igg( extstyleigg( extstyleigg( extstyleigg( extstyleigg( extstyleigg( k=1 igg(5k + 8 $\bigg)$\bigg)$\bigg)$\bigg)$\bigg)$\bigg(5k + 8$\bigg(5k + 8$\bigg)$\bigg)$\bigg)$\bigg)$\bigg(5k + 8$\bigg)$\bigg)

where the lower limit of the summation is $k=1$ and the upper limit of the summation is $k=9$.

Sum Elements Definition

The sum consists of the following linear expression with respect to $k$:

• $5k + 8$ where:
  • $5k$ is a linear function that scales $k$ by a factor of 5,
  • $8$ is a constant that shifts the result by 8.

Expanding the Summation

We can express the sum as:

$extstyle\bigg(5(1) + 8$ + $5(2) + 8$ + $5(3) + 8$ + … + $5(9) + 8$\bigg)

Therefore, we rewrite the individual components of the sum as follows:

• When k=1: 5(1) + 8 = 5 + 8 = 13
• When k=2: 5(2) + 8 = 10 + 8 = 18
• When k=3: 5(3) + 8 = 15 + 8 = 23
• When k=4: 5(4) + 8 = 20 + 8 = 28
• When k=5: 5(5) + 8 = 25 + 8 = 33
• When k=6: 5(6) + 8 = 30 + 8 = 38
• When k=7: 5(7) + 8 = 35 + 8 = 43
• When k=8: 5(8) + 8 = 40 + 8 = 48
• When k=9: 5(9) + 8 = 45 + 8 = 53

So, we can compile all computed values for each iteration of k:
ext{Sum} = 13 + 18 + 23 + 28 + 33 + 38 + 43 + 48 + 53

Final Evaluation of the Sum

To compute the overall sum:

1. Start adding the values together:

   • Sum of first three terms: 13 + 18 + 23 = 54
   • Sum of the next three terms: 28 + 33 + 38 = 99
   • Sum of the last three terms: 43 + 48 + 53 = 144

2. Now sum these partial results:
   54 + 99 + 144 = 297

Thus, the evaluated sum is:
ext{Total Sum} = 297

Conclusion

Therefore, the evaluated sum of the series defined by the expression is:

oxed{297}$$