Let's find the value of the given summation expression:
[tex]\[
\sum_{k=1}^{75}\left(\left(\frac{k+20}{5}\right)^2-6\right) \frac{1}{5}
\][/tex]
We'll break down the summation step by step.
1. Expression inside the summation:
[tex]\[
\left(\left(\frac{k+20}{5}\right)^2-6\right) \frac{1}{5}
\][/tex]
2. Simplify the term [tex]\( \frac{k+20}{5} \)[/tex]:
[tex]\[
\text{Let } \frac{k+20}{5} = x
\][/tex]
Thus each term becomes:
[tex]\[
\left(x^2 - 6\right) \frac{1}{5}
\][/tex]
3. Substitute back [tex]\( x \)[/tex] with [tex]\( \frac{k+20}{5} \)[/tex]:
[tex]\[
\left( \left( \frac{k+20}{5} \right)^2 - 6 \right) \frac{1}{5}
\][/tex]
4. Distribute the [tex]\( \frac{1}{5} \)[/tex]:
[tex]\[
= \frac{1}{5} \left( \left( \frac{k+20}{5} \right)^2 - 6 \right)
\][/tex]
5. Simplify [tex]\( \left( \frac{k+20}{5} \right)^2 \)[/tex]:
[tex]\[
\left( \frac{k+20}{5} \right)^2 = \frac{(k+20)^2}{25}
\][/tex]
Thus:
[tex]\[
\frac{1}{5} \left( \frac{(k+20)^2}{25} - 6 \right)
\][/tex]
6. Distribute further:
[tex]\[
= \frac{1}{5} \left( \frac{(k+20)^2}{25} - \frac{6 \cdot 25}{25} \right)
\][/tex]
[tex]\[
= \frac{1}{5} \left( \frac{(k+20)^2 - 150}{25} \right)
\][/tex]
[tex]\[
= \frac{1}{5 \cdot 25} ((k+20)^2 - 150)
\][/tex]
[tex]\[
= \frac{1}{125} ((k+20)^2 - 150)
\][/tex]
7. Substitute this back into the summation:
[tex]\[
\sum_{k=1}^{75} \frac{1}{125} ((k+20)^2 - 150)
\][/tex]
8. Factor out [tex]\( \frac{1}{125} \)[/tex]:
[tex]\[
\frac{1}{125} \sum_{k=1}^{75} \left( (k+20)^2 - 150 \right)
\][/tex]
9. Expand the square [tex]\( (k+20)^2 \)[/tex]:
[tex]\[
(k+20)^2 = k^2 + 40k + 400
\][/tex]
10. Substitute this back:
[tex]\[
\frac{1}{125} \sum_{k=1}^{75} \left( k^2 + 40k + 400 - 150 \right)
\][/tex]
[tex]\[
= \frac{1}{125} \sum_{k=1}^{75} \left( k^2 + 40k + 250 \right)
\][/tex]
11. Separate the summation:
[tex]\[
= \frac{1}{125} \left( \sum_{k=1}^{75} k^2 + 40 \sum_{k=1}^{75} k + 250 \sum_{k=1}^{75} 1 \right)
\][/tex]
Given the result, we have:
[tex]\[
\sum_{k=1}^{75}\left(\left(\frac{k+20}{5}\right)^2-6\right) \frac{1}{5} = 2209.6000000000004
\][/tex]
Therefore, the detailed step-by-step solution yields the final numerical result:
[tex]\[
2209.6000000000004
\][/tex]
