To solve the sum [tex]\(\sum_{n=1}^{24}(2n - 3)\)[/tex], let's break it down step by step.
1. Express the sum explicitly:
[tex]\[
\sum_{n=1}^{24}(2n - 3) = (2 \cdot 1 - 3) + (2 \cdot 2 - 3) + (2 \cdot 3 - 3) + \cdots + (2 \cdot 24 - 3)
\][/tex]
2. Separate the sum into two individual sums:
[tex]\[
\sum_{n=1}^{24}(2n - 3) = \sum_{n=1}^{24} 2n - \sum_{n=1}^{24} 3
\][/tex]
3. Factor out constants from the sums:
[tex]\[
\sum_{n=1}^{24}(2n - 3) = 2 \sum_{n=1}^{24} n - 3 \sum_{n=1}^{24} 1
\][/tex]
4. Compute each sum separately:
The first sum, [tex]\(\sum_{n=1}^{24} n\)[/tex], is the sum of the first 24 natural numbers. The formula for the sum of the first [tex]\(N\)[/tex] natural numbers is:
[tex]\[
\sum_{n=1}^{N} n = \frac{N(N+1)}{2}
\][/tex]
For [tex]\(N=24\)[/tex],
[tex]\[
\sum_{n=1}^{24} n = \frac{24 \cdot 25}{2} = 300
\][/tex]
The second sum, [tex]\(\sum_{n=1}^{24} 1\)[/tex], is simply adding the number 1 twenty-four times, which gives:
[tex]\[
\sum_{n=1}^{24} 1 = 24
\][/tex]
5. Substitute back into the expression:
[tex]\[
\sum_{n=1}^{24}(2n - 3) = 2 \cdot 300 - 3 \cdot 24
\][/tex]
6. Perform the arithmetic operations:
[tex]\[
2 \cdot 300 = 600
\][/tex]
[tex]\[
3 \cdot 24 = 72
\][/tex]
[tex]\[
600 - 72 = 528
\][/tex]
Thus, the sum [tex]\(\sum_{n=1}^{24}(2n - 3)\)[/tex] is equal to [tex]\(528\)[/tex].
So, the answer is [tex]\(528\)[/tex].
