To solve the summation [tex]\(\sum_{n=1}^{26}(2n + 5)\)[/tex], follow these steps:
1. Separate the summation into two parts:
[tex]\[
\sum_{n=1}^{26} (2n + 5) = \sum_{n=1}^{26} 2n + \sum_{n=1}^{26} 5
\][/tex]
2. Evaluate the first series [tex]\(\sum_{n=1}^{26} 2n\)[/tex]:
- Factor out the constant 2:
[tex]\[
\sum_{n=1}^{26} 2n = 2 \sum_{n=1}^{26} n
\][/tex]
- Use the formula for the sum of the first [tex]\(N\)[/tex] natural numbers [tex]\(\sum_{n=1}^{N} n = \frac{N(N+1)}{2}\)[/tex], where [tex]\(N = 26\)[/tex]:
[tex]\[
\sum_{n=1}^{26} n = \frac{26 \cdot 27}{2} = 351
\][/tex]
- Multiply by 2:
[tex]\[
2 \sum_{n=1}^{26} n = 2 \cdot 351 = 702
\][/tex]
3. Evaluate the second series [tex]\(\sum_{n=1}^{26} 5\)[/tex]:
- Since the sum of a constant [tex]\(k\)[/tex] over [tex]\(N\)[/tex] terms is [tex]\(k \cdot N\)[/tex]:
[tex]\[
\sum_{n=1}^{26} 5 = 5 \cdot 26 = 130
\][/tex]
4. Add the results of the two series together:
[tex]\[
\sum_{n=1}^{26} (2n + 5) = 702 + 130 = 832
\][/tex]
Therefore, the value of the summation [tex]\(\sum_{n=1}^{26}(2n + 5)\)[/tex] is [tex]\(\boxed{832}\)[/tex].
