Just as we studied special types of sequences, we will look at special types of series. Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is the common difference. The sum of the terms of an arithmetic sequence is called an arithmetic series.
formula for the partial sum of an arithmetic series
The sum, [latex]S_n[/latex], of the first [latex]n[/latex] terms of an arithmetic sequence is
If we add these two expressions for the sum of the first [latex]n[/latex] terms of an arithmetic series, we can derive a formula for the sum of the first [latex]n[/latex] terms of any arithmetic series.
This is generally referred to as the partial sum of the series.
How To: Given terms of an arithmetic series, find the partial sum
Identify [latex]{a}_{1}[/latex] and [latex]{a}_{n}[/latex].
Determine [latex]n[/latex].
Substitute values for [latex]{a}_{1},{a}_{n}[/latex], and [latex]n[/latex] into the formula [latex]{S}_{n}=\dfrac{n\left({a}_{1}+{a}_{n}\right)}{2}[/latex].
Simplify to find [latex]{S}_{n}[/latex].
Find the sum of the first [latex]n[/latex] terms of an arithmetic sequence.
Find the sum of the first [latex]30[/latex] terms of the arithmetic sequence:
[latex]7, 10, 13, 13, 19,...[/latex]
[latex]\begin{align*} \text{To find the 30th term, use the formula with } a_1 = 7, \, d = 3, \text{ and } n = 30.\end{align*}[/latex]
[latex]a_n = a_1 + (n - 1)d[/latex]
[latex]\begin{align*} \text{Substitute} \quad & a_{30} = 7 + (30 - 1)(3) \end{align*}[/latex]
[latex]\begin{align*} \text{Simplify} \quad & a_{30} = 7 + (29)(8) \\ & a_{30} = 7 + 232 \\ & a_{30} = 239 \end{align*}[/latex]
[latex]\begin{align*} \text{To find } S_{30} \text{ use the formula with } a_1 = 7, \, a_{30} = 239, \text{ and } n = 30. \end{align*}[/latex]
[latex]S_n = \frac{n}{2}(a_1 + a_n)[/latex]
[latex]\begin{align*} \text{Substitute and simplify} \quad & S_{30} = \frac{30}{2} (7 + 239) \\ & S_{30} = 15(246) \\ & S_{30} = 3690 \end{align*}[/latex]
Find the sum of the first [latex]50[/latex] terms of the arithmetic sequence whose general term is [latex]a_n = 2n-5[/latex].
[latex]\begin{align*} \text{To find the sum of the first 50 terms, we start by finding } a_1 \text{ and } a_{50}. \\ \text{The general term is given by } a_n &= 2n - 5. \\ \text{First, find the first term } a_1: & \\ a_1 &= 2(1) - 5 \\ a_1 &= 2 - 5 \\ a_1 &= -3 \\ \text{Next, find the 50th term } a_{50}: & \\ a_{50} &= 2(50) - 5 \\ a_{50} &= 100 - 5 \\ a_{50} &= 95 \\ \text{Now, use the sum formula for an arithmetic sequence:} & \\ S_n &= \frac{n}{2} \left(a_1 + a_n\right) \\ S_{50} &= \frac{50}{2} \left(-3 + 95\right) \\ S_{50} &= 25 \times 92 \\ S_{50} &= 2300 \end{align*}[/latex]
Find the sum
[latex]\sum_{i=1}^{30} (6i-4)[/latex]
[latex]\begin{align*} \text{To find the sum } \sum_{i=1}^{30} (6i-4), \text{ first identify the sequence terms.} & \\ \text{The first term is:} & \\ a_1 &= 6(1) - 4 \\ a_1 &= 6 - 4 \\ a_1 &= 2 \\ \text{The 30th term is:} & \\ a_{30} &= 6(30) - 4 \\ a_{30} &= 180 - 4 \\ a_{30} &= 176 \\ \text{Now, use the sum formula:} & \\ S_n &= \frac{n}{2} \left(a_1 + a_n\right) \\ S_{30} &= \frac{30}{2} \left(2 + 176\right) \\ S_{30} &= 15 \times 178 \\ S_{30} &= 2670 \end{align*}[/latex]