Let's solve the given sum step-by-step:
The given expression to be summed from [tex]\( k = 1 \)[/tex] to [tex]\( k = 75 \)[/tex] is:
[tex]$ \sum_{k=1}^{75} \left(\left(\frac{2k + 39}{10}\right)^2 - 6\right) \frac{1}{10} $[/tex]
First, let's break down the expression inside the summation:
1. Simplify the expression inside the parentheses:
The expression inside the parentheses is given by:
[tex]$ \left(\frac{2k + 39}{10}\right)^2 - 6 $[/tex]
Let's simplify [tex]\(\left(\frac{2k + 39}{10}\right)^2\)[/tex]:
[tex]\[
\left(\frac{2k + 39}{10}\right)^2 = \left(\frac{2k + 39}{10}\right) \times \left(\frac{2k + 39}{10}\right) = \frac{(2k + 39)^2}{100}
\][/tex]
So our expression simplifies to:
[tex]$ \frac{(2k + 39)^2}{100} - 6 $[/tex]
2. Multiply the entire expression by [tex]\(\frac{1}{10}\)[/tex]:
Now, we take:
[tex]$ \left(\frac{(2k + 39)^2}{100} - 6\right) \frac{1}{10} $[/tex]
Distribute [tex]\(\frac{1}{10}\)[/tex]:
[tex]$ \frac{(2k + 39)^2}{1000} - \frac{6}{10} $[/tex]
Simplify [tex]\(\frac{6}{10}\)[/tex]:
[tex]$ \frac{6}{10} = 0.6 $[/tex]
So the expression we need to sum from [tex]\( k = 1 \)[/tex] to [tex]\( k = 75 \)[/tex] is:
[tex]$ \frac{(2k + 39)^2}{1000} - 0.6 $[/tex]
3. Write the full summation:
Now, substitute the simplified expression into the summation:
[tex]$ \sum_{k=1}^{75} \left(\frac{(2k + 39)^2}{1000} - 0.6\right) $[/tex]
4. Split the summation into two parts:
Using the property of summations, [tex]\(\sum (a - b) = \sum a - \sum b\)[/tex]:
[tex]$ \sum_{k=1}^{75} \frac{(2k + 39)^2}{1000} - \sum_{k=1}^{75} 0.6 $[/tex]
5. Calculate each sum separately:
- First sum:
[tex]$ \sum_{k=1}^{75} \frac{(2k + 39)^2}{1000} $[/tex]
This sum requires evaluating [tex]\(\frac{(2k + 39)^2}{1000}\)[/tex] for each [tex]\(k\)[/tex] from 1 to 75 and then summing them up.
- Second sum:
Since 0.6 is a constant, we multiply it by the number of terms (75):
[tex]$ \sum_{k=1}^{75} 0.6 = 0.6 \times 75 = 45 $[/tex]
6. Combine the results:
Now, combine the results from the two parts:
[tex]$ \left(\sum_{k=1}^{75} \frac{(2k + 39)^2}{1000}\right) - 45 $[/tex]
After evaluating the first sum and subtracting 45:
Thus, the result of the given summation is:
[tex]$ 11279.75 $[/tex]
