Reasoning about values in an arithmetic sequence | Maths

Starter quiz

In sequences an $n^\text{th}$ term rule is a __________-to-term rule.

term

position ✓

solution
Match these sequences to their respective $n^\text{th}$ term rules.

$3,6,9,12,15, ...$

⇔

$3n$ ✓

$5,8,11,14,17, ...$

⇔

$3n+2$ ✓

$5,10,15,20,25, ...$

⇔

$5n$ ✓

$7,12,17,22,27, ...$

⇔

$5n+2$ ✓

$1,6,11,16,21, ...$

⇔

$5n-4$ ✓

$1,4,7,10,13, ...$

⇔

$3n-2$ ✓
What is the $15^\text{th}$ term of the sequence $4n-12$ ?

'48' ✓
What is the $15^\text{th}$ term of the sequence $12-4n$ ?

'-48' ✓
Solve the equation $6n-35=331$ $n=$ ______

'61' ✓
Solve the equation $35-6n=-487$ $n=$ ______

'87' ✓

Exit quiz

If $4832$ is in the sequence $5n-13$ then $5n-13=4832$ must have a __________.

solution

positive solution

positive integer solution ✓

positive non-integer solution

negative integer solution
What is the $n^\text{th}$ term rule for the sequence $-14,-11,-8,-5,-2,1, ...$ ?

$-17-3n$

$-14-3n$

$17-3n$

$3n-14$

$3n-17$ ✓
Is $406$ in the sequence $-14,-11,-8,-5,-2,1, ...$ ?

Yes - the $n^\text{th}$ term rule is $3n-17$ ✓

Yes - the $n^\text{th}$ term rule is $-3n-14$

No - the $n^\text{th}$ term rule is $3n-17$

No - the $n^\text{th}$ term rule is $-3n-14$
How might you show that $1138$ is not in the sequence $7n-19$ ?

By explaining that $1138$ is even but all positive terms of $7n-19$ are odd.

By calculating the first few terms and 'counting on'.

By forming and solving an equation and showing that $n$ is not an integer. ✓

By forming and solving an equation and showing that $n$ is a positive integer.
What is true of the arithmetic sequence $24, 29, 34, 39, 44, ...$ ?

$3428$ will be a term.

$3429$ will be a term. ✓

The terms will always be odd.

$3824$ will be a term. ✓

$-344$ will be a term.
$707$ is the $53^\text{rd}$ term and $720$ the $54^\text{th}$ term of an arithmetic sequence. Which of these terms are in the same sequence?

$538$ ✓

$550$

$577$ ✓

$824$ ✓

$864$

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Lesson Details

Key learning points

It is possible to generate the sequence and therefore show whether a value appears in the sequence.
It is possible to use your knowledge of multiples to reason whether a value appears in a given sequence.

Common misconception

As long as the equation can be solved, the value will be in the sequence.

For the value to be in the sequence, the value for $n$ must be a positive integer. If it is not, then the value is not in the sequence as the term number must be a positive integer.

Keywords

Arithmetic sequence - An arithmetic (or linear) sequence is a sequence where the difference between successive terms is a constant.
N^th term - The nth term of a sequence is the position of a term in a sequence where n stands for the term number.