Sure, let's evaluate the given summation step-by-step.
We need to find the sum of the series:
[tex]\[
\sum_{n=1}^{45} (4n + 9)
\][/tex]
First, we can break this sum into two separate sums:
[tex]\[
\sum_{n=1}^{45} (4n + 9) = \sum_{n=1}^{45} 4n + \sum_{n=1}^{45} 9
\][/tex]
### Step 1: Evaluate [tex]\(\sum_{n=1}^{45} 4n\)[/tex]
We can factor out the constant 4 from the summation:
[tex]\[
\sum_{n=1}^{45} 4n = 4 \sum_{n=1}^{45} n
\][/tex]
The sum of the first [tex]\(N\)[/tex] natural numbers is given by the formula:
[tex]\[
\sum_{n=1}^{N} n = \frac{N (N + 1)}{2}
\][/tex]
For [tex]\(N = 45\)[/tex]:
[tex]\[
\sum_{n=1}^{45} n = \frac{45 \cdot 46}{2} = 1035
\][/tex]
So:
[tex]\[
\sum_{n=1}^{45} 4n = 4 \cdot 1035 = 4140
\][/tex]
### Step 2: Evaluate [tex]\(\sum_{n=1}^{45} 9\)[/tex]
Since 9 is a constant, this sum simply becomes:
[tex]\[
\sum_{n=1}^{45} 9 = 9 \cdot 45 = 405
\][/tex]
### Step 3: Combine the results
Now, we add the results from Step 1 and Step 2:
[tex]\[
\sum_{n=1}^{45} (4n + 9) = 4140 + 405 = 4545
\][/tex]
Thus, the evaluated sum is:
[tex]\[
\boxed{4545}
\][/tex]
