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$\\var{n[0]*index[0]}$, $\\var{n[0]*(index[0]+1)}$, $\\var{n[0]*(index[0]+2)}$, $\\var{n[0]*(index[0]+3)}$

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Common Difference = [[0]]

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$\\var{n[1]*(index[1])}$, $?$, $\\var{n[1]*(index[1]+2)}$, $?$, $\\var{n[1]*(index[1]+4)}$

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Common Difference = [[0]]

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$\\var{n[2]*(index[2])}$, $?$, $?$, $\\var{n[2]*(index[2]+3)}$, $?$, $?$, $\\var{n[2]*(index[2]+6)}$

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Common Difference = [[0]]

Given sequences with missing terms, find the common difference between terms.

"}, "tags": ["Arithmetic Sequences", "Arithmetic sequences", "arithmetic sequences", "common difference", "sequences", "taxonomy"], "variables": {"n": {"templateType": "anything", "description": "", "definition": "shuffle(3..9)[0..3]", "name": "n", "group": "Ungrouped variables"}, "index": {"templateType": "anything", "description": "", "definition": "shuffle(5..15 except 10)[0..3]", "name": "index", "group": "Ungrouped variables"}}, "rulesets": {}, "extensions": [], "functions": {}, "ungrouped_variables": ["n", "index"], "statement": "

Find the common differences of the following arithmetic sequences. Some of the terms are missing.

In an arithmetic sequence, the difference between two adjecent terms is always the same.

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Call the common difference $d$, and the first term in the sequence $a_0$. Then the sequence goes as follows:

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\$a_0, \\; a_0+d, \\; a_0+2d, \\; a_0+3d, \\; \\ldots \$

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The difference between a term in the sequence and the term $n$ places along is $n \\times d$.

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#### a)

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The difference between the first two terms is $\\var{n[0]*(index[0]+1)} - \\var{n[0]*index[0]} = \\var{n[0]}$.

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So the common difference is $\\var{n[0]}$.

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#### b)

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We're not given any adjacent terms of this sequence, but we are given some terms two palces apart.

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\\begin{align}
2d &= \\var{n[1]*(index[1]+2)} - \\var{n[1]*index[1]} \\\\
&= \\var{2*n[1]} \\\\
d &= \\var{n[1]}
\\end{align}

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The common difference is $\\var{n[1]}$.

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#### c)

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Again we're not given any adjacent terms of this sequence, but we have two terms three places apart.

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\\begin{align}
3d &= \\var{n[2]*(index[2]+3)} - \\var{n[2]*index[2]} \\\\
&= \\var{3*n[2]} \\\\
d &= \\var{n[2]}
\\end{align}

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The common difference is $\\var{n[2]}$.

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