Proving that a sequence is arithmetic

Sequences, grade 12 (4 units) Lesson 8 of 20Learning

Proving that a sequence is arithmetic

A general proof

A sequence is arithmetic when the difference $\displaystyle a_{n+1}-a_n$ is the same number for every $\displaystyle n$ . Checking a few terms is no proof: the next term might not fit.
The proof below shows that the sequence $\displaystyle {a_n}={7-3n}$ is arithmetic:

$\displaystyle {a_{n+1}}={7-3(n+1)}={4-3n}$ .
$\displaystyle {a_{n+1}-a_n}={(4-3n)-(7-3n)}={-3}$ .
The difference is $\displaystyle -3$ for every $\displaystyle n$ , so the sequence is arithmetic, and $\displaystyle {d}={-3}$ .

A counterexample

To show that a sequence is not arithmetic, one example is enough: two pairs of neighbouring terms with different differences.
In the sequence $\displaystyle {a_n}={n^2+1}$ : $\displaystyle {a_1}={2}$ , $\displaystyle {a_2}={5}$ , $\displaystyle {a_3}={10}$ .

$\displaystyle {a_2-a_1}={3}$ , but $\displaystyle {a_3-a_2}={5}$ .
The differences are not equal, so the sequence is not arithmetic.

From the recursion rule

The rule $\displaystyle {a_{n+1}}={a_n+5}$ says at once that the difference is 5 for every $\displaystyle n$ : the sequence is arithmetic.
In the rule $\displaystyle {a_1}={12}$ , $\displaystyle {a_{n+1}}={3a_n-2}$ : $\displaystyle {a_2}={34}$ , $\displaystyle {a_3}={100}$ . $\displaystyle {a_2-a_1}={22}$ and $\displaystyle {a_3-a_2}={66}$ , so the sequence is not arithmetic.

From the sum to the term:

\displaystyle {S_n-S_{n-1}}={a_n}

$\displaystyle {S_n}={a_1+\ldots+a_{n-1}+a_n}$ , and $\displaystyle {S_{n-1}}={a_1+\ldots+a_{n-1}}$ . Their difference is the last term:

\displaystyle S_n-S_{n-1}

\displaystyle =

\displaystyle a_n

\displaystyle n\geq2

and $\displaystyle a_1$ is $\displaystyle S_1$ – the sum of one term.
In the example below $\displaystyle {S_n}={n^2+4n}$ is given. We find $\displaystyle a_n$ , and prove that the sequence is arithmetic:

$\displaystyle {S_{n-1}}={(n-1)^2+4(n-1)}={n^2+2n-3}$ .
$\displaystyle {a_n}={S_n-S_{n-1}}={(n^2+4n)-(n^2+2n-3)}={2n+3}$ , for every $\displaystyle n\geq2$ .
$\displaystyle {a_1}={S_1}={1+4}={5}$ , and also $\displaystyle {2\cdot1+3}={5}$ : the formula $\displaystyle {a_n}={2n+3}$ holds for every $\displaystyle n$ .
$\displaystyle {a_{n+1}-a_n}={2}$ , so the sequence is arithmetic, and $\displaystyle {d}={2}$ .

Exercise 1

The sequence: $\displaystyle {a_n}={4n+9}$ .

Work out $\displaystyle a_{n+1}-a_n$ .

/ – 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

Is the sequence arithmetic? If so, what is its difference?

Exercise 2

The sequence: $\displaystyle {a_n}={2n^2}$ .
Show by a counterexample that the sequence is not arithmetic.

Exercise 3

The sum of the first $\displaystyle n$ terms of a sequence: $\displaystyle {S_n}={3n^2-n}$ .

Find $\displaystyle a_n$ for every $\displaystyle n\geq2$ .

/ – 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

Calculate $\displaystyle a_1$ , and check that it fits the formula too.

/ – 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 arithmetic? If so, what is its difference?

Exercise 4

The sum of the first $\displaystyle n$ terms of a sequence: $\displaystyle {S_n}={n^2+n+1}$ .

Find $\displaystyle a_n$ for every $\displaystyle n\geq2$ .

/ – 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

Calculate $\displaystyle a_1$ . Is the sequence arithmetic? Explain.

\displaystyle a_1

/ – 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

Formulas: sequences, grade 12 (4 units)

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