Alright, let's solve the given summation step-by-step.
We need to evaluate the summation:
[tex]\[
\sum_{k=1}^{75} \left( \left( \frac{k+19}{5} \right)^2 - 6 \right) \frac{1}{5}
\][/tex]
First, let's break it down into simpler parts. We can rewrite the sum as:
[tex]\[
\sum_{k=1}^{75} \left( \left( \frac{k+19}{5} \right)^2 \cdot \frac{1}{5} - 6 \cdot \frac{1}{5} \right)
\][/tex]
Simplify the constants outside and inside the summation:
[tex]\[
\sum_{k=1}^{75} \left( \frac{1}{5} \left( \frac{k+19}{5} \right)^2 - \frac{6}{5} \right)
\][/tex]
Further, we can split this into two separate sums:
[tex]\[
\frac{1}{5} \sum_{k=1}^{75} \left( \frac{k+19}{5} \right)^2 - \frac{6}{5} \sum_{k=1}^{75} 1
\][/tex]
Let's handle each part separately.
### First Part
[tex]\[
\frac{1}{5} \sum_{k=1}^{75} \left( \frac{k+19}{5} \right)^2
\][/tex]
Notice that [tex]\( \left( \frac{k+19}{5} \right)^2 \)[/tex] involves squaring the fraction inside:
[tex]\[
\left( \frac{k+19}{5} \right)^2 = \frac{(k+19)^2}{25}
\][/tex]
So, the first part becomes:
[tex]\[
\frac{1}{5} \sum_{k=1}^{75} \frac{(k+19)^2}{25} = \frac{1}{5 \cdot 25} \sum_{k=1}^{75} (k+19)^2 = \frac{1}{125} \sum_{k=1}^{75} (k+19)^2
\][/tex]
### Second Part
[tex]\(
\frac{6}{5} \sum_{k=1}^{75} 1 = \frac{6}{5} \cdot 75 = 90
\)[/tex]
### Combining the Results
We have:
[tex]\[
\text{First Part} - \text{Second Part}
\][/tex]
The first part involves a more complicated sum, but we have:
[tex]\[
\frac{1}{125} \sum_{k=1}^{75} (k + 19)^2
\][/tex]
And then, subtract:
[tex]\[
- 90
\][/tex]
The final result is approximately:
[tex]\[
2140.6000000000004
\][/tex]
So, the final answer is:
[tex]\[
\boxed{2140.6000000000004}
\][/tex]
