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Evaluate the following summation:
[tex]\[ \sum_{k=1}^{75}\left(\left(\frac{k+19}{5}\right)^2-6\right) \frac{1}{5} \][/tex]


Sagot :

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]
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