Partial Sums Of An Arithmetic Sequence - Milefoot

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Partial Sums of an Arithmetic Sequence

A finite number of terms of an arithmetic sequence can be added to find their sum. Since an arithmetic sequence always has an unbounded long-term behavior, we are always restricted to adding a finite number of terms.

Specific Numerical Results

Consider the sum $8+13+18+23+\ldots+273$. We quickly recognize that the terms have a common difference of 5, and this is therefore the sum of an arithmetic sequence whose explicit formula is $a_n=5n+3$. Thus the sequence of partial sums is defined by $s_n=\sum\limits_{k=1}^n (5k+3)$, for some value of $n$. Solving the equation $5n+3=273$, we determine that 273 is the 54th term of the sequence.

Using a little ingenuity, we might proceed as follows.

\begin{align} & \phantom2 s = \phantom{28}8+\phantom{2}13+\phantom{2}18+\phantom{2}23+\ldots+273 \\ & \underline{\phantom2 s = 273+268+263+258+\ldots+\phantom{28}8} \\ & 2s = 281+281+281+281+\ldots+281 \\ & \phantom2 s = \frac{281(54)}{2} = 7587 \end{align}

In a very similar fashion, it can be shown that every arithmetic sequence has the $n$th partial sum $s_n = \dfrac{n}{2}(a_1+a_n)$.

An alternative approach is to use the properties of sums.

\begin{equation*} \sum_{k=1}^{54} (5k+3) = 5 \sum_{k=1}^{54} k + \sum_{k=1}^{54} 3 = (5)\frac{54(55)}{2}+54(3) = 7587 \end{equation*}

Specific Partial Sum Formulas

Either approach can be used to create a formula for the sequence of partial sums $s_n=\sum\limits_{k=1}^n (5k+3)$, simply by leaving $n$ as a variable. This gives us the following explicit formulas for the sequence of partial sums.

• $s_n=\dfrac{n}{2}(a_1+a_n)=\dfrac{n}{2}(8+5n+3)=\dfrac{n(5n+11)}{2}$
• $\sum\limits_{k=1}^n (5k+3)=5\sum\limits_{k=1}^n k+\sum\limits_{k=1}^n 3=(5)\dfrac{n(n+1)}{2}+3n=\dfrac{n(5n+11)}{2}$

Applications

The arithmetic sequence is used in a variety of applications. Examples include:

• The seats in an auditorium are arranged in a semicircular pattern. There are 30 rows, the first row has 24 seats, and each succeeding row has an additional 2 seats. The explicit formula for the sequence will be $a_n=2n+22$, the last row has $a_{30}=2(30)+22=82$ seats, and there are $s_n=\dfrac{30}{2}(24+82)=1590$ seats in the auditorium.
• Campbell County's grain production was 150 million bushels in 1990, and since then has been increasing by 3 million bushels per year. The explicit formula (in millions) is $a_n=3n+147$, the county produced $a_{21}=3(21)+147=210$ million bushels in the year 2010, and a total of $s_{21}=\dfrac{21}{2}(150+210)=3780$ million bushels during those 21 years.