To solve the problem, we will write the given sum [tex]\(\sum_{n=2}^5(2n+3)\)[/tex] in expanded form and then find its value by adding the terms. Here is the detailed step-by-step process:
1. Identify the given sum expression:
[tex]\[
\sum_{n=2}^5(2n+3)
\][/tex]
2. Substitute the values of [tex]\(n\)[/tex] from 2 to 5 into the expression [tex]\(2n + 3\)[/tex]:
- For [tex]\(n = 2\)[/tex]:
[tex]\[
2(2) + 3 = 4 + 3 = 7
\][/tex]
- For [tex]\(n = 3\)[/tex]:
[tex]\[
2(3) + 3 = 6 + 3 = 9
\][/tex]
- For [tex]\(n = 4\)[/tex]:
[tex]\[
2(4) + 3 = 8 + 3 = 11
\][/tex]
- For [tex]\(n = 5\)[/tex]:
[tex]\[
2(5) + 3 = 10 + 3 = 13
\][/tex]
3. Write the expanded form using the calculated values:
[tex]\[
\sum_{n=2}^5(2n+3) = 7 + 9 + 11 + 13
\][/tex]
4. Calculate the sum of the expanded form:
[tex]\[
7 + 9 + 11 + 13
\][/tex]
- First, add [tex]\(7\)[/tex] and [tex]\(9\)[/tex]:
[tex]\[
7 + 9 = 16
\][/tex]
- Next, add [tex]\(11\)[/tex] to [tex]\(16\)[/tex]:
[tex]\[
16 + 11 = 27
\][/tex]
- Finally, add [tex]\(13\)[/tex] to [tex]\(27\)[/tex]:
[tex]\[
27 + 13 = 40
\][/tex]
Therefore, the expanded form of the given sum is [tex]\([7, 9, 11, 13]\)[/tex] and the value of the sum is [tex]\(40\)[/tex].
