Let's solve the given summation step-by-step:
The expression given is:
[tex]\[
\sum_{n=1}^5 (2n + 1)
\][/tex]
This notation means we need to find the sum of \(2n + 1\) for \(n\) ranging from 1 to 5. Let's calculate what each term in the series is:
For \(n = 1\):
[tex]\[
2(1) + 1 = 2 + 1 = 3
\][/tex]
For \(n = 2\):
[tex]\[
2(2) + 1 = 4 + 1 = 5
\][/tex]
For \(n = 3\):
[tex]\[
2(3) + 1 = 6 + 1 = 7
\][/tex]
For \(n = 4\):
[tex]\[
2(4) + 1 = 8 + 1 = 9
\][/tex]
For \(n = 5\):
[tex]\[
2(5) + 1 = 10 + 1 = 11
\][/tex]
Now, we add all these terms together:
[tex]\[
3 + 5 + 7 + 9 + 11
\][/tex]
Let's sum these numbers step-by-step:
First, add 3 and 5:
[tex]\[
3 + 5 = 8
\][/tex]
Next, add 8 and 7:
[tex]\[
8 + 7 = 15
\][/tex]
Next, add 15 and 9:
[tex]\[
15 + 9 = 24
\][/tex]
Finally, add 24 and 11:
[tex]\[
24 + 11 = 35
\][/tex]
So the sum of the series from \(n = 1\) to \(n = 5\) for the expression \(2n + 1\) is:
[tex]\[
\sum_{n=1}^5 (2n + 1) = 35
\][/tex]
