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Both of these sequences are linear, or arithmetic, sequences. To find formulas for these sequences we need to identify their first terms and common differences.

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a)

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We can use a table and the equation: \\[a_n=a_1+(n-1)d\\text{;}\\] with $a_1$ being the first term and $d$ the common difference; to find the formula for the sequence:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$n$123
$a_n$$\\pmb{\\var{m[1]*2}}$$\\var{m[1]*3}$$\\var{m[1]*4}$
First differences$\\pmb{\\var{m[1]}}$$\\pmb{\\var{m[1]}}$
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The first term and common difference have been highlighted in bold and we can use these to form the formula for the sequence.

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\\[ \\begin{align}n^{\\text{th}}\\text{term} &=a_1+(n-1)d\\\\ &=\\var{m[1]*2}+(n-1)\\times\\var{m[1]}\\\\&=\\var{m[1]}n + \\var{m[1]}\\text{.}\\end{align}\\]

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b)

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We can use a table and the equation: \\[a_n=a_1+(n-1)d\\text{;}\\] with $a_1$ being the first term and $d$ the common difference; to find the formula for the sequence:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$n$123
$a_n$$\\pmb{\\var{m[2]*8+2}}$$\\simplify{{m[2]}*7+2}$$\\simplify{{m[2]}*6+2}$
First differences$\\pmb{\\var{-m[2]}}$$\\pmb{\\var{-m[2]}}$
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The first term and common difference have been highlighted in bold and we can use these to form the formula for the sequence. 

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\\[ \\begin{align}n^{\\text{th}}\\text{term} &=a_1+(n-1)d\\\\ &=\\var{m[2]*8+2}+(n-1)\\times\\var{-m[2]}\\\\&=-\\var{m[2]}n + \\var{m[2]*9+2}\\text{.}\\end{align}\\]

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$\\var{m[1]*2}, \\var{m[1]*3}, \\var{m[1]*4}...$

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$n^{th}$ term = [[0]]

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The arithmetic formula is

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\\[a_n = a_1 + (n-1)d, \\]

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where
$a_n$ - The n$^{th}$ term in an arithmetic sequence 
$a_1$ - The $1^{st}$ term in an arithmetic sequence 
$n$ - Term number
$d$ - The common difference.

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For this arithmetic sequence, what is $a_1$?

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What is $d$?

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$\\var{m[2]*8+2}, \\var{m[2]*7+2}, \\var{m[2]*6+2}...$

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$n^\\text{th}$ term = [[0]]

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The arithmetic formula is

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\\[a_n = a_1 + (n-1)d, \\]

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where
$a_n$ - The n$^{th}$ term in an arithmetic sequence 
$a_1$ - The $1^{st}$ term in an arithmetic sequence 
$n$ - Term number 
$d$ - The common difference.

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For this arithmetic sequence, what is $a_1$?

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What is $d$?

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Question assessing the students understanding of linear sequences.

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Students are assessed on their ability to find the common difference and first term in a linear sequence and then find the nth term of the sequence using the arithmetic formula.

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A linear sequence is a series of numbers that either increases or decreases by a constant amount at each step.

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Find formulas for the $n^{\\text{th}}$ term for each of the following linear sequences, where the values for $n=1\\text{,}2\\text{,}3$ are given:

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