// Numbas version: exam_results_page_options {"name": "Identifying different types of sequences", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"questions": [{"type": "question", "name": "Identifying different types of sequences", "variablesTest": {"maxRuns": 100, "condition": ""}, "variable_groups": [{"variables": ["a1", "b1", "c2"], "name": "seq' eqn' question c)"}], "parts": [{"type": "m_n_x", "showCorrectAnswer": true, "minAnswers": 0, "displayType": "radiogroup", "variableReplacements": [], "shuffleAnswers": false, "maxAnswers": 0, "shuffleChoices": false, "matrix": [["1", 0], [0, "1"], [0, "1"]], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "choices": ["

ii) $\\var{m[3]*5}, \\var{m[3]*6}, \\var{m[3]*7}, \\var{m[3]*8}...$

", "

iii) $\\var{m[2]*10}, \\var{m[2]*11}, \\var{m[2]*12}...$

", "

iv) $\\var{b[4]*1^2+n[4]*1+c[4]}, \\var{b[4]*2^2+n[4]*2+c[4]}, \\var{b[4]*3^2+n[4]*3+c[4]}, \\var{b[4]*4^2+n[4]*4+c[4]}...$

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", "

Linear

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For each of these sequences state whether they are linear or quadratic.

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i) $\\var{c[0]}, \\var{c[0]^2}, \\var{c[0]^3}, \\var{c[0]^4}\\ldots$

", "

ii) $\\var{c[2]*c[3]}, \\var{c[2]*c[3]^2}, \\var{c[2]*c[3]^3}, \\var{c[2]*c[3]^4}\\ldots$

", "

iii) $\\var{c[1]*5}, \\var{c[1]*6}, \\var{c[1]*7}, \\var{c[1]*8}, \\ldots$

", "

iv) $\\var{c[2]*8}, \\var{c[2]*9}, \\var{c[2]*10}, \\var{c[2]*11} \\ldots$

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Arithmetic

", "

Geometric

"], "prompt": "

Which of these sequences are arithmetic and which are geometric?

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The $n^{th}$ term of the triangle sequence is

\n

\$\\frac{n(n+1)}{2}\\text{.}\$

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Write out the next three terms in the triangle sequence

\n

$1, 3, 6, 10,$ [[0]], [[1]], [[2]]

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Find the $\\var{ci[0]}$th term of the sequence with formula:

\n

\$a_n=\\frac{\\var{a1}n(n+\\var{b1})}{\\var{c2}}\$

\n

$a_{\\var{ci[0]}}=$ [[0]]

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Write out the next three terms of the following sequences.

\n

i)

\n

$1, 4, 9, 16, 25,$ [[0]], [[1]], [[2]]

\n

ii)

\n

$1, 8, 27, 64,$ [[3]], [[4]], [[5]]

#### a)

\n

A linear sequence is a sequence with a constant common difference whilst a quadratic sequence has a difference which increases or decreases by a constant value. We can calculate the differences between terms by constructing tables. These tables can then be used to decide whether the sequence is linear or quadratic.

\n

i)

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $n$ $1$ $2$ $3$ $4$ $a_n$ $\\var{b[2]+n[2]+c[2]}$ $\\var{b[2]*2^2+n[2]*2+c[2]}$ $\\var{b[2]*3^2+n[2]*3+c[2]}$ $\\var{b[2]*4^2+n[2]*4+c[2]}$ Differences between terms $\\var{b[2]*2^2+n[2]*2+c[2]-b[2]*1^2-n[2]*1-c[2]}$ $\\var{b[2]*3^2+n[2]*3+c[2]-b[2]*2^2-n[2]*2-c[2]}$ $\\var{b[2]*4^2+n[2]*4+c[2]-b[2]*3^2-n[2]*3-c[2]}$
\n

If we look at the differences from this pattern we see they increase by $\\var{2*b[2]}$. As the original differences in the sequence increase by a constant the sequence is quadratic.

\n

\n

ii)

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $n$ $1$ $2$ $3$ $4$ $a_n$ $\\var{m[3]*5}$ $\\var{m[3]*6}$ $\\var{m[3]*7}$ $\\var{m[3]*8}$ Differences between terms $\\var{m[3]}$ $\\var{m[3]}$ $\\var{m[3]}$
\n

This sequence increase by $\\var{m[3]}$ therefore is linear.

\n

\n

iii)

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $n$ $1$ $2$ $3$ $a_n$ $\\var{m[2]*10}$ $\\var{m[2]*11}$ $\\var{m[2]*12}$ Differences between terms $\\var{m[2]}$ $\\var{m[2]}$
\n

\n

iv)

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $n$ $1$ $2$ $3$ $4$ $a_n$ $\\var{b[4]*1^2+n[4]*1+c[4]}$ $\\var{b[4]*2^2+n[4]*2+c[4]}$ $\\var{b[4]*3^2+n[4]*3+c[4]}$ $\\var{b[4]*4^2+n[4]*4+c[4]}$ Differences between terms $\\var{3*b[4]+n[4]}$ $\\var{5*b[4]+n[4]}$ $\\var{7*b[4]+n[4]}$
\n

\n

If we look at the differences from this pattern we see they increase by $\\var{2*b[4]}$. As the original differences in the sequence increase by a constant the sequence is quadratic.

\n

#### b)

\n

Arithmetic sequences can be identified by the fact that they have a common difference, whereas geometric sequences have a common ratio.

\n

i)

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $n$ $1$ $2$ $3$ $4$ $a_n$ $\\var{c[0]}$ $\\var{c[0]^2}$ $\\var{c[0]^3}$ $\\var{c[0]^4}$ Differences between terms $\\var{c[0]^2-c[0]}$ $\\var{c[0]^3-c[0]^2}$ $\\var{c[0]^4-c[0]^3}$
\n

\n

This sequence has a common ratio of $\\var{c[0]}$ therefore the sequence is geometric.

\n

ii)

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $n$ $1$ $2$ $3$ $4$ $a_n$ $\\var{c[2]*c[3]}$ $\\var{c[2]*c[3]^2}$ $\\var{c[2]*c[3]^3}$ $\\var{c[2]*c[3]^4}$ Differences between terms $\\var{c[2]*c[3]^2-c[2]*c[3]}$ $\\var{c[2]*c[3]^3-c[2]*c[3]^2}$ $\\var{c[2]*c[3]^4-c[2]*c[3]^3}$
\n

This sequence has a common ratio of $\\var{c[3]}$ therefore the sequence is geometric.

\n

iii)

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $n$ $1$ $2$ $3$ $4$ $a_n$ $\\\\var{c[1]*5}$ $\\var{c[1]*6}$ $\\var{c[1]*7}$ $\\var{c[1]*8}$ Differences between terms $\\var{c[1]}$ $\\var{c[1]}$ $\\var{c[1]}$
\n

Each term in this sequence has a common difference of $\\var{c[1]}$ therefore the sequence is arithmetic.

\n

iv)

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $n$ $1$ $2$ $3$ $4$ $a_n$ $\\var{c[2]*8}$ $\\var{c[2]*9}$ $\\var{c[2]*10}$ $\\var{c[2]*11}$ Differences between terms $\\var{c[2]}$ $\\var{c[2]}$ $\\var{c[2]}$
\n

Each term in this sequence has a common difference of $\\var{c[2]}$ therefore the sequence is arithmetic.

\n

\n

#### c)

\n

We can use a table to identify the triangle sequence by it's charecteristic of the difference between terms increasing by $1$ with each term.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $n$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $a_n$ $1$ $3$ $6$ $10$ $a_5$ $a_6$ $a_7$ Differences between terms $2$ $3$ $4$ $5$ $6$ $7$
\n

We can then use this pattern to continue the sequence and identify the next three terms ($a_5$, $a_6$ and $a_7$).

\n

\$a_5=a_4+5=15\$

\n

\$a_6=a_5+6=21\$

\n

\$a_7=a_6+7=28\$

\n

OR

\n

We can use the formula for the $n^{th}$ term of the triangle sequence

\n

\$\\frac{n(n+1)}{2}\\text{.}\$

\n

to find the $5th, 6th\\; \\text{and}\\; 7th$ terms.

\n

\\\begin{align} \\frac{5(5+1)}{2}&=15\\\\ \\frac{6(6+1)}{2}&=21\\\\ \\frac{7(7+1)}{2}&=28 \\end{align}\

\n

#### d)

\n

\n

To find the answer to this question we must substitute our value for n ($\\var{ci[0]}$) into the formula for the sequence we have already been given:

\n

\$a_n=\\frac{\\var{a1}n(n+\\var{b1})}{\\var{c2}}\\text{.}\$

\n

Therefore:

\n

\\\begin{align} a_{\\var{ci[0]}}&=\\frac{\\var{a1}n(n+\\var{b1})}{\\var{c2}}\\\\ &=\\frac{\\var{a1}\\times\\var{ci[0]}(\\var{ci[0]}+\\var{b1})}{\\var{c2}}\\\\ &=\\simplify{{{a1}*{ci[0]}*({ci[0]}+{b1})}/{c2}} \\end{align} \

\n

\n

#### e)

\n

i)

\n

We can analyse the sequence by using a table to visualise each term with the difference between itself and the previous term.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $n$ $1$ $2$ $3$ $4$ $5$ $a_n$ $1$ $4$ $9$ $16$ $25$ Differences between terms $3$ $5$ $7$ $9$
\n

The we observe that the difference between terms increases with each term, and by carefully observing the sequence, we may realise that  this is a square sequence meaning each term has been squared to obtain its value.

\n

\\\begin{align} 1^2&=1\\\\ 2^2&=4\\\\ 3^2&=9\\\\ 4^2&=16\\\\ 5^2&=25 \\end{align}\

\n

Therefore to obtain the next three values we have to square the values

\n

\\\begin{align} 6^2&=36\\\\ 7^2&=49\\\\ 8^2&=64\\text{.} \\end{align}\

\n

ii)

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $n$ $1$ $2$ $3$ $4$ $a_n$ $1$ $8$ $27$ $64$ Differences between terms $7$ $21$ $37$
\n

The we observe that the difference between terms increases with each term, and by carefully observing the sequence, we may realise that  this is a cubic sequence meaning each term has been cubed to obtain its value.

\n

\\\begin{align} 1^3&=1\\\\ 2^3&=8\\\\ 3^3&=27\\\\ 4^3&=64 \\end{align}\

\n

Therefore to obtain the next three values we have to cube the values

\n

\\\begin{align} 5^3&=125\\\\ 6^3&=216\\\\ 7^3&=343\\text{.} \\end{align}\

\n

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A sequence is a list of numbers that follow a pattern. Different types of sequences follow different patterns, for example some increase by a constant amount term to term and others follow a rule of multiplying the previous term by a constant find the next term.