Evaluation of the Sum

To evaluate the sum given by the series $\sum_{k=1}^{n} (5k + 8)$, we first identify its components. The sum starts from $k=1$ and encompasses terms of the form $5k + 8$. Here, we demonstrate the step-by-step evaluation process.

Step 1: Substituting Values

We calculate the initial terms of the sum for a few values of $k$:

• For $k=1$:
  $5(1) + 8 = 5 + 8 = 13$
• For $k=2$:
  $5(2) + 8 = 10 + 8 = 18$
• For $k=3$:
  $5(3) + 8 = 15 + 8 = 23$

Therefore, the first three terms of the sum are $13, 18,$ and $23$.

Step 2: Summing the Terms

We consider summing these terms, and we can write the sum as:

• $S = (5(1) + 8) + (5(2) + 8) + (5(3) + 8) + \dots + (5(n) + 8)$

Step 3: Factoring out Common Terms

Factoring out the common terms from the sum, we can rewrite the sum as follows:
$S = \sum_{k=1}^{n} (5k + 8) = \sum_{k=1}^{n} 5k + \sum_{k=1}^{n} 8$

Now, we can separately calculate each part.

• The first part $\sum_{k=1}^{n} 5k$ is 5 times the sum of the first n natural numbers:
  $\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}$
    Hence,
  $\sum_{k=1}^{n} 5k = 5 \cdot \frac{n(n + 1)}{2} = \frac{5n(n + 1)}{2}$
• The second sum $\sum_{k=1}^{n} 8$ is simply 8 times the number of terms, which equals:
  $8n$

Step 4: Final Expression

Combining these two results gives us:
$S = \frac{5n(n + 1)}{2} + 8n$

Step 5: Simplifying the Expression

To simplify our expression, we can express it as a single fraction:
"$S = \frac{5n(n + 1)}{2} + \frac{16n}{2} = \frac{5n(n + 1) + 16n}{2}$
  Simplifying this further, we have:
$= \frac{5n^2 + 5n + 16n}{2}$
$= \frac{5n^2 + 21n}{2}$

Conclusion

The evaluation of the sum $\sum_{k=1}^{n} (5k + 8)$ results in the final expression:
$S = \frac{5n^2 + 21n}{2}$
This formula allows for the computation of the sum for any specified upper limit $n$.