To find the sum of the arithmetic sequence given, [tex]\(5, -1, -7, -13, \ldots, -427\)[/tex], follow these detailed steps:
1. Find the common difference ([tex]\(d\)[/tex]) of the sequence:
The common difference is the difference between any two consecutive terms. Typically, we use the first two terms to find this.
[tex]\[
d = \text{second term} - \text{first term} = -1 - 5 = -6
\][/tex]
2. Find the number of terms ([tex]\(n\)[/tex]) in the sequence:
To find the number of terms, we use the [tex]\(n\)[/tex]-th term formula for an arithmetic sequence:
[tex]\[
a_n = a_1 + (n-1)d
\][/tex]
Here, the [tex]\(n\)[/tex]-th term is given as -427, the first term [tex]\(a_1\)[/tex] is 5, and the common difference [tex]\(d\)[/tex] is -6. Plug these values into the formula and solve for [tex]\(n\)[/tex]:
[tex]\[
-427 = 5 + (n-1)(-6)
\][/tex]
Rearrange the equation to solve for [tex]\(n\)[/tex]:
[tex]\[
-427 = 5 - 6(n-1)
\][/tex]
Simplify further:
[tex]\[
-427 = 5 - 6n + 6
\][/tex]
[tex]\[
-427 = 11 - 6n
\][/tex]
Move the constants to the other side of the equation:
[tex]\[
-438 = -6n
\][/tex]
Solving for [tex]\(n\)[/tex]:
[tex]\[
n = \frac{-438}{-6} = 73
\][/tex]
3. Find the sum ([tex]\(S_n\)[/tex]) of the arithmetic sequence:
The sum of an arithmetic series can be found using the formula:
[tex]\[
S_n = \frac{n}{2} (a_1 + a_n)
\][/tex]
Here we have [tex]\(n = 73\)[/tex], the first term [tex]\(a_1 = 5\)[/tex], and the [tex]\(n\)[/tex]-th term [tex]\(a_n = -427\)[/tex]. Plug these values into the formula:
[tex]\[
S_{73} = \frac{73}{2} (5 + (-427))
\][/tex]
Simplify the expression:
[tex]\[
S_{73} = \frac{73}{2} (5 - 427) = \frac{73}{2} (-422) = 73 \cdot (-211) = -15403
\][/tex]
Therefore, the sum of the arithmetic sequence [tex]\(5, -1, -7, -13, \ldots, -427\)[/tex] is [tex]\(-15403\)[/tex].
