To write the series [tex]\( 5 + 6 + 7 + \cdots + 12 \)[/tex] using summation notation, let’s follow these steps:
1. Recognize that this is an arithmetic series where each term increases by 1.
2. Identify the first term [tex]\(a\)[/tex] and the last term [tex]\(l\)[/tex]:
- First term ([tex]\(a\)[/tex]) = 5
- Last term ([tex]\(l\)[/tex]) = 12
- Common difference ([tex]\(d\)[/tex]) = 1
3. Determine the number of terms [tex]\(n\)[/tex] in the series. To find [tex]\(n\)[/tex], use the formula for the [tex]\(n\)[/tex]-th term of an arithmetic series:
[tex]\[
l = a + (n-1)d
\][/tex]
Plugging in the known values:
[tex]\[
12 = 5 + (n-1) \cdot 1
\][/tex]
Simplify the equation to find [tex]\(n\)[/tex]:
[tex]\[
12 = 5 + (n-1)
\][/tex]
[tex]\[
12 - 5 = n - 1
\][/tex]
[tex]\[
7 = n - 1
\][/tex]
[tex]\[
n = 8
\][/tex]
So, there are 8 terms in the series.
4. Write the series in summation notation. Since it's an arithmetic series starting at 5 and ending at 12 with a common difference of 1, the [tex]\(k\)[/tex]-th term can be written as:
[tex]\[
a_k = 5 + (k-1) \cdot 1 = 5 + k - 1 = k + 4
\][/tex]
Thus, the series in summation notation will be:
[tex]\[
\sum_{k=1}^{8} (k + 4)
\][/tex]
Therefore, the series [tex]\( 5 + 6 + 7 + \cdots + 12 \)[/tex] can be written in summation notation as:
[tex]\[
\sum_{k=1}^{8} (k + 4)
\][/tex]
