Answer :
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]
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]
