The arithmetic sequence

Sequences, grade 12 (4 units) Lesson 4 of 20Learning

The arithmetic sequence

A constant difference

An arithmetic sequence is a sequence in which each term equals the term before it plus a fixed number, $\displaystyle d$ – the common difference:

\displaystyle a_{n+1}=a_n+d

$\displaystyle 12, 14, 16, 18, \ldots$ – $\displaystyle {d}={2}$ ;
$\displaystyle 30, 25, 20, 15, \ldots$ – $\displaystyle {d}={-5}$ : the difference is negative, and the terms get smaller;
$\displaystyle 7, 7, 7, \ldots$ – $\displaystyle {d}={0}$ .

$\displaystyle d$ is found from any two neighbouring terms: $\displaystyle {d}={a_{n+1}-a_n}$ .

Four ways to show an arithmetic sequence

A cinema has 12 seats in the first row, and each row has 2 seats more than the row before it.

In a table:

$\displaystyle n$	1	2	3	4	5
$\displaystyle a_n$	12	14	16	18	20

By a recursion rule: $\displaystyle {a_1}={12}$ , $\displaystyle {a_{n+1}}={a_n+2}$ .
On a graph – the points $\displaystyle (n, a_n)$ :

\displaystyle (1,12)

\displaystyle (2,14)

\displaystyle (3,16)

\displaystyle (4,18)

\displaystyle (5,20)

\displaystyle y=2x+10

The points lie on a line (the dashed one): with each step to the right $\displaystyle n$ grows by 1, and the term rises by $\displaystyle {d}={2}$ . The graph of an arithmetic sequence is points on a line whose slope is $\displaystyle d$ – a discrete graph of a linear function.

The general term

From $\displaystyle a_1$ to $\displaystyle a_n$ we add $\displaystyle d$ , $\displaystyle n-1$ times:

$\displaystyle {a_2}={a_1+d}$ ;
$\displaystyle {a_3}={a_1+2d}$ ;
$\displaystyle {a_4}={a_1+3d}$ .

And so for every place:

\displaystyle a_n

\displaystyle =

\displaystyle a_1+(n-1)d

In the cinema: row 30 has $\displaystyle {a_{30}}={12+29\cdot2}={70}$ seats.

Increasing, decreasing, constant

The sign of $\displaystyle d$ decides how the sequence behaves:

$\displaystyle d>0$ – the sequence is increasing: each term is greater than the one before it;
$\displaystyle d<0$ – the sequence is decreasing: each term is less than the one before it;
$\displaystyle {d}={0}$ – the sequence is constant.

Exercise 1

Which of the sequences are arithmetic? For an arithmetic one, write $\displaystyle d$ .

$\displaystyle 4, 9, 14, 19, \ldots$
$\displaystyle 2, 4, 8, 16, \ldots$
$\displaystyle 10, 6.5, 3, -0.5, \ldots$

Exercise 2

The arithmetic sequence: $\displaystyle 7, 11, 15, \ldots$
Calculate:

\displaystyle a_{25}

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle a_{100}

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

The graph shows the first five terms of an arithmetic sequence.

Write on the board a function whose graph passes through all the points.

\displaystyle (1,5)

\displaystyle (2,8)

\displaystyle (3,11)

\displaystyle (4,14)

\displaystyle (5,17)

What is the slope of the line, and what is the difference of the sequence?

The slope

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle d

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

In an arithmetic sequence $\displaystyle {a_1}={40}$ and $\displaystyle {d}={-3}$ .

Calculate $\displaystyle a_{15}$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Is the sequence increasing or decreasing? Explain.

Formulas: sequences, grade 12 (4 units)

Open as a page

A sequence and the sum of its first terms

The term from the sum – $\displaystyle S_n$ the sum of the first $\displaystyle n$ terms
$\displaystyle S_n-S_{n-1}$ $\displaystyle =$ $\displaystyle a_n$ $\displaystyle n\geq2$
The first term is the sum of one term
$\displaystyle a_1$ $\displaystyle =$ $\displaystyle S_1$

The arithmetic sequence

The recursion rule – $\displaystyle d$ the common difference
$\displaystyle a_{n+1}=a_n+d$
The general term
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle a_1+(n-1)d$
The arithmetic mean of the neighbours
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle \frac{a_{n-1}+a_{n+1}}{2}$ $\displaystyle n\geq2$
The sum of the first $\displaystyle n$ terms
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{n(a_1+a_n)}{2}$
The sum without the last term
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{n(2a_1+(n-1)d)}{2}$

The geometric sequence

The recursion rule – $\displaystyle q$ the common ratio
$\displaystyle a_{n+1}=a_n\cdot q$
The general term
$\displaystyle a_n$ $\displaystyle =$ $\displaystyle a_1\cdot q^{n-1}$
The geometric mean: a term’s square is the product of its neighbours
$\displaystyle a_n^2$ $\displaystyle =$ $\displaystyle a_{n-1}\cdot a_{n+1}$ $\displaystyle n\geq2$
The sum of the first $\displaystyle n$ terms
$\displaystyle S_n$ $\displaystyle =$ $\displaystyle \frac{a_1(q^n-1)}{q-1}$ $\displaystyle q\neq1$

The infinite geometric sequence

The sum of the infinite sequence
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a_1}{1-q}$ $\displaystyle -1<q<1,\ q\neq0$