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:

extstyleigg(5(1) + 8 + $5(2) + 8$ + $5(3) + 8$ + … + $5(9) + 8$\bigg)$</p> <p>Therefore, we rewrite the individual components of the sum as follows:</p> <ul> <li>When$k=1: 5(1) + 8 = 5 + 8 = 13$</li> <li>When$k=2: 5(2) + 8 = 10 + 8 = 18$</li> <li>When$k=3: 5(3) + 8 = 15 + 8 = 23$</li> <li>When$k=4: 5(4) + 8 = 20 + 8 = 28$</li> <li>When$k=5: 5(5) + 8 = 25 + 8 = 33$</li> <li>When$k=6: 5(6) + 8 = 30 + 8 = 38$</li> <li>When$k=7: 5(7) + 8 = 35 + 8 = 43$</li> <li>When$k=8: 5(8) + 8 = 40 + 8 = 48$</li> <li>When$k=9: 5(9) + 8 = 45 + 8 = 53$</li> </ul> <p>So, we can compile all computed values for each iteration of$k$:<br />$ ext{Sum} = 13 + 18 + 23 + 28 + 33 + 38 + 43 + 48 + 53$</p> <h4 id="finalevaluationofthesum">Final Evaluation of the Sum</h4> <p>To compute the overall sum:</p> <ol> <li><p>Start adding the values together:</p> <ul> <li>Sum of first three terms:$13 + 18 + 23 = 54$</li> <li>Sum of the next three terms:$28 + 33 + 38 = 99$</li> <li>Sum of the last three terms:$43 + 48 + 53 = 144$</li></ul></li> <li><p>Now sum these partial results:<br />$54 + 99 + 144 = 297$</p></li> </ol> <p>Thus, the evaluated sum is:<br />$ ext{Total Sum} = 297$</p> <h5 id="conclusion">Conclusion</h5> <p>Therefore, the evaluated sum of the series defined by the expression is:</p> <p>$oxed{297}$$