The first term, the difference and the arithmetic mean

Sequences, grade 12 (4 units) Lesson 6 of 20Learning

The first term, the difference and the arithmetic mean

Two terms given

In an arithmetic sequence $\displaystyle {a_4}={17}$ and $\displaystyle {a_9}={32}$ .
The solution below writes each term with $\displaystyle a_1$ and $\displaystyle d$ , and solves a system of two equations:

\begin{cases}a_1+3d=17\\a_1+8d=32\end{cases}

Subtract the first equation from the second: $\displaystyle {5d}={15}$ , so $\displaystyle {d}={3}$ .
Substitute in the first: $\displaystyle {a_1+9}={17}$ , so $\displaystyle {a_1}={8}$ .
The sequence: $\displaystyle 8, 11, 14, 17, \ldots$ – and indeed $\displaystyle {a_9}={8+8\cdot3}={32}$ .

Steps of

\displaystyle d

From $\displaystyle a_4$ to $\displaystyle a_9$ there are 5 steps of $\displaystyle d$ , so $\displaystyle {a_9-a_4}={5d}$ : $\displaystyle {32-17}={15}={5d}$ , and $\displaystyle {d}={3}$ without a system.
In general, between $\displaystyle a_k$ and $\displaystyle a_m$ there are $\displaystyle m-k$ steps: $\displaystyle {a_m-a_k}={(m-k)d}$ .

The arithmetic mean

In an arithmetic sequence each term is the arithmetic mean of the two terms on either side of it – hence the name “arithmetic sequence”:

\displaystyle a_n

\displaystyle =

\displaystyle \frac{a_{n-1}+a_{n+1}}{2}

\displaystyle n\geq2

The reason: $\displaystyle {a_{n-1}}={a_n-d}$ and $\displaystyle {a_{n+1}}={a_n+d}$ , so $\displaystyle {a_{n-1}+a_{n+1}}={2a_n}$ .
For example, in the sequence $\displaystyle 8, 11, 14$ : $\displaystyle {\frac{8+14}{2}}={11}$ .

Example: three neighbouring terms

The numbers $\displaystyle x-1$ , $\displaystyle 2x+1$ , $\displaystyle 4x-3$ (in this order) are three neighbouring terms of an arithmetic sequence.
The middle term is the mean of the other two, so $\displaystyle {2(2x+1)}={(x-1)+(4x-3)}$ :

$\displaystyle {4x+2}={5x-4}$ , so $\displaystyle {x}={6}$ .
The numbers: $\displaystyle 5, 13, 21$ – the difference is 8.

Exercise 1

In an arithmetic sequence $\displaystyle {a_3}={11}$ and $\displaystyle {a_8}={26}$ .

Find $\displaystyle a_1$ and $\displaystyle d$ : write a system of equations and solve it.

Calculate $\displaystyle a_{20}$ .

/ – 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

Exercise 2

In an arithmetic sequence, the sum of the first and the third term is 20.
The fifth term is 22.

Find $\displaystyle a_1$ and $\displaystyle d$ .

Calculate $\displaystyle a_{10}$ .

/ – 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 numbers $\displaystyle x+2$ , $\displaystyle 3x$ , $\displaystyle 4x+1$ (in this order) are three neighbouring terms of an arithmetic sequence.

Find $\displaystyle x$ .

Write the three numbers and the difference.

Exercise 4

A sequence is defined by the recursion rule:

\begin{cases}a_1=12\\a_{n+1}=3a_n-2\end{cases}

Calculate $\displaystyle a_2$ and $\displaystyle a_3$ .

\displaystyle a_2

/ – 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_3

/ – 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

The numbers $\displaystyle a_1-2$ , $\displaystyle a_2-4$ , $\displaystyle a_3-x$ are the first three terms of an arithmetic sequence.
Find $\displaystyle x$ .

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
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a_1}{1-q}$ $\displaystyle -1<q<1,\ q\neq0$