Finite arithmetic series

# Arithmetic series

Walk through a guided practice where you'll start by finding a simple sum and end by evaluating finite arithmetic series.

Let's start with an addition problem.

### Find the sum of $1 + 3 + 5 + 7 + 9$1, plus, 3, plus, 5, plus, 7, plus, 9.

**Your answer should be**- an integer, like $6$6
- a
*simplified proper*fraction, like $3/5$3, slash, 5 - a
*simplified improper*fraction, like $7/4$7, slash, 4 - a mixed number, like $1\ 3/4$1, space, 3, slash, 4
- an
*exact*decimal, like $0.75$0, point, 75 - a multiple of pi, like $12\ \text{pi}$12, space, p, i or $2/3\ \text{pi}$2, slash, 3, space, p, i

Awesome! You just found the sum of a small arithmetic series. It only had $5$5 terms. But, what if it had one million terms? We'd want a formula for sure. Thankfully, we've already learned of such a formula.

**Identify the formula for the sum of an arithmetic series.**

### Formula for arithmetic series

The sum $S_n$S, start subscript, n, end subscript of a finite arithmetic series is

$~~~~~S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$space, space, space, space, space, S, start subscript, n, end subscript, equals, start fraction, left parenthesis, a, start subscript, 1, end subscript, plus, a, start subscript, n, end subscript, right parenthesis, divided by, 2, end fraction, dot, n

where $a_1$a, start subscript, 1, end subscript is the first term, $a_n$a, start subscript, n, end subscript is the last term, and $n$n is the number of terms.

Sweet! So you remember the formula. Now let's make sure we remember how to apply it.

**Choose the answer that shows the formula correctly used to find the sum you found.**

The first term $a_1$a, start subscript, 1, end subscript is $\greenD1$start color greenD, 1, end color greenD. The last term $a_n$a, start subscript, n, end subscript is $\goldD9$start color goldD, 9, end color goldD. The number of terms $n$n is $\blueD5$start color blueD, 5, end color blueD.

$S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$S, start subscript, n, end subscript, equals, start fraction, left parenthesis, a, start subscript, 1, end subscript, plus, a, start subscript, n, end subscript, right parenthesis, divided by, 2, end fraction, dot, n

$\phantom{S_n = }\dfrac{(\greenD1 + \goldD9)}{2} \cdot \blueD5$empty space, start fraction, left parenthesis, start color greenD, 1, end color greenD, plus, start color goldD, 9, end color goldD, right parenthesis, divided by, 2, end fraction, dot, start color blueD, 5, end color blueD

This expression equals $25$25 as it should!

Alright, so we're feeling good so far. Let's try to use the formula to find the sum of an arithmetic series that would be tedious to calculate by hand.

## Consider the series $3 + 5 + 7 + ... + 401$3, plus, 5, plus, 7, plus, point, point, point, plus, 401.

**Find the values of $a_1$a, start subscript, 1, end subscript and $a_n$a, start subscript, n, end subscript for this series.**

$a_1 =$a, start subscript, 1, end subscript, equals

**Your answer should be**- an integer, like $6$6
- a
*simplified proper*fraction, like $3/5$3, slash, 5 - a
*simplified improper*fraction, like $7/4$7, slash, 4 - a mixed number, like $1\ 3/4$1, space, 3, slash, 4
- an
*exact*decimal, like $0.75$0, point, 75 - a multiple of pi, like $12\ \text{pi}$12, space, p, i or $2/3\ \text{pi}$2, slash, 3, space, p, i

$a_n =$a, start subscript, n, end subscript, equals

**Your answer should be**- an integer, like $6$6
- a
*simplified proper*fraction, like $3/5$3, slash, 5 - a
*simplified improper*fraction, like $7/4$7, slash, 4 - a mixed number, like $1\ 3/4$1, space, 3, slash, 4
- an
*exact*decimal, like $0.75$0, point, 75 - a multiple of pi, like $12\ \text{pi}$12, space, p, i or $2/3\ \text{pi}$2, slash, 3, space, p, i

The first term $a_1$a, start subscript, 1, end subscript is $\greenD3$start color greenD, 3, end color greenD.

The last term $a_n$a, start subscript, n, end subscript is $\goldD{401}$start color goldD, 401, end color goldD.

**Find the value of $n$n for this series.**

$n =$n, equals

**Your answer should be**- an integer, like $6$6
- a
*simplified proper*fraction, like $3/5$3, slash, 5 - a
*simplified improper*fraction, like $7/4$7, slash, 4 - a mixed number, like $1\ 3/4$1, space, 3, slash, 4
- an
*exact*decimal, like $0.75$0, point, 75 - a multiple of pi, like $12\ \text{pi}$12, space, p, i or $2/3\ \text{pi}$2, slash, 3, space, p, i

The sequence increases by $401 - 3 = 398$401, minus, 3, equals, 398 from the first term to the last term. Because the sequence increases by $2$2 each time, it takes $\dfrac{398}{2} = 199$start fraction, 398, divided by, 2, end fraction, equals, 199 terms to get from the first term to the last term. We still need to count the first term, so there are $199 + 1 = \blueD{200}$199, plus, 1, equals, start color blueD, 200, end color blueD terms in the sequence.

In other words, $n = \blueD{200}$n, equals, start color blueD, 200, end color blueD.

**Find the sum of $3 + 5 + 7 + ... + 401$3, plus, 5, plus, 7, plus, point, point, point, plus, 401**

**Your answer should be**- an integer, like $6$6
- a
*simplified proper*fraction, like $3/5$3, slash, 5 - a
*simplified improper*fraction, like $7/4$7, slash, 4 - a mixed number, like $1\ 3/4$1, space, 3, slash, 4
- an
*exact*decimal, like $0.75$0, point, 75 - a multiple of pi, like $12\ \text{pi}$12, space, p, i or $2/3\ \text{pi}$2, slash, 3, space, p, i

The first term $a_1$a, start subscript, 1, end subscript is $\greenD3$start color greenD, 3, end color greenD. The last term $a_n$a, start subscript, n, end subscript is $\goldD{401}$start color goldD, 401, end color goldD. The number of terms $n$n is $\blueD{200}$start color blueD, 200, end color blueD.

$\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\\\\\
S_{\blueD{200}}&= \dfrac {\left(\greenD3 + \goldD{401} \right)}{2} \cdot \blueD{200} \\\\\\\\
S_{{200}} &= \dfrac{404}{2} \cdot 200\\\\\\\\
S_{{200}} &= 202(200) \\\\\\\\
S_{{200}} &= 40{,}400\end{aligned}$

Wow! Okay, looks like you've got this.

## Try it yourself

Problem 1

**Find the sum.**

$11 + 20 + 29 + ... + 4052=$11, plus, 20, plus, 29, plus, point, point, point, plus, 4052, equals

**Your answer should be**- an integer, like $6$6
- a
*simplified proper*fraction, like $3/5$3, slash, 5 - a
*simplified improper*fraction, like $7/4$7, slash, 4 - a mixed number, like $1\ 3/4$1, space, 3, slash, 4
- an
*exact*decimal, like $0.75$0, point, 75 - a multiple of pi, like $12\ \text{pi}$12, space, p, i or $2/3\ \text{pi}$2, slash, 3, space, p, i

The first term $a_1$a, start subscript, 1, end subscript is $\greenD{11}$start color greenD, 11, end color greenD. The last term $a_n$a, start subscript, n, end subscript is $\goldD{4052}$start color goldD, 4052, end color goldD. The number of terms $n$n is $\blueD{450}$start color blueD, 450, end color blueD.

$\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\\\\\
S_{\blueD{450}}&= \dfrac {\left(\greenD{11} + \goldD{4052} \right)}{2} \cdot \blueD{450} \\\\\\\\
S_{{450}} &= \dfrac{4063}{2} \cdot 450\\\\\\\\
S_{{450}} &= 2031.5(450) \\\\\\\\
S_{{450}} &= 914{,}175\end{aligned}$

Nice! Try another one!

Problem 2

**Find the sum.**

$10 + (-1) + (-12) + ... + (-10{,}979)=$10, plus, left parenthesis, minus, 1, right parenthesis, plus, left parenthesis, minus, 12, right parenthesis, plus, point, point, point, plus, left parenthesis, minus, 10, comma, 979, right parenthesis, equals

**Your answer should be**- an integer, like $6$6
- a
*simplified proper*fraction, like $3/5$3, slash, 5 - a
*simplified improper*fraction, like $7/4$7, slash, 4 - a mixed number, like $1\ 3/4$1, space, 3, slash, 4
- an
*exact*decimal, like $0.75$0, point, 75 - a multiple of pi, like $12\ \text{pi}$12, space, p, i or $2/3\ \text{pi}$2, slash, 3, space, p, i

The first term $a_1$a, start subscript, 1, end subscript is $\greenD{10}$start color greenD, 10, end color greenD. The last term $a_n$a, start subscript, n, end subscript is $\goldD{-10{,}979}$start color goldD, minus, 10, comma, 979, end color goldD. The number of terms $n$n is $\blueD{1000}$start color blueD, 1000, end color blueD.

$\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\\\\\
S_{\blueD{1000}}&= \dfrac {\left(\greenD{10} + \goldD{-10{,}979} \right)}{2} \cdot \blueD{1000} \\\\\\\\
S_{{1000}} &= \dfrac{-10{,}969}{2} \cdot 1000\\\\\\\\
S_{{1000}} &= -5484.5(1000) \\\\\\\\
S_{{1000}} &=-5{,}484{,}500\end{aligned}$

Finite arithmetic series