Sequences and Series
Arithmetic Sequences
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Question 1 of 6
1. Question
Given the sequence `1+5+9…`, find:`(i)` The common difference`(ii)` The `11th` term-
`(i)` `d=` (4)`(ii)` `U_11=` (41)
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Help VideoCommon Difference Formula
$$d=U_2-U_1=U_3-U_2$$General Rule of an Arithmetic Sequence
$$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$`(i)` Finding the common differenceFirst, identify the consecutive values$$U_\color{#CC0000}{1}$$ `=` `1` $$U_\color{#CC0000}{2}$$ `=` `5` $$U_\color{#CC0000}{3}$$ `=` `9` Next, use the formula to solve for the common difference`d` `=` `U_2-U_1` `=` `5-1` Substitute known values `=` `4` `(ii)` Finding the `10th` termSubstitute the known values to the general rule`\text(Number of terms)``[n]` `=` `11` `\text(First term)``[a]` `=` `1` `\text(Common Difference)``[d]` `=` `4` $$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$U_{\color{#9a00c7}{11}}$$ `=` $$\color{#e65021}{1}+[(\color{#9a00c7}{11}-1)\color{#00880A}{4}]$$ Substitute known values `=` `1+[(10)4]` Evaluate `=` `41` `(i) d=4``(ii) U_11=41` -
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Question 2 of 6
2. Question
Given the sequence `87,83,79,75…`, find:`(i)` If the expression is an arithmetic sequence`(ii)` The general rule form`(iii)` The `38th` term`(iv)` If `-132` is part of the sequenceFor part `i` and `iv`, write `Y` for yes and `N` for no-
`(i)` (Y, y)`(ii)` `U_n=` (91-4n)`(iii)` `U_38=` (-61)`(iv)` (N, n)
Hint
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Help VideoCommon Difference Formula
$$d=U_2-U_1=U_3-U_2$$General Rule of an Arithmetic Sequence
$$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$`(i)` Finding if the expression is an arithmetic sequenceFirst, identify the consecutive values$$U_\color{#CC0000}{1}$$ `=` `1` $$U_\color{#CC0000}{2}$$ `=` `5` $$U_\color{#CC0000}{3}$$ `=` `9` Next, use the formula to solve for the common difference and check if it is consistent with the consecutive values`d` `=` `U_2-U_1` `=` `83-87` Substitute known values `=` `-4` `d` `=` `U_3-U_2` `=` `79-83` Substitute known values `=` `-4` `d` `=` `U_4-U_3` `=` `75-79` Substitute known values `=` `-4` The common difference is consistent. Therefore, the expression is an arithmetic sequence.`(ii)` Finding the general rule formSubstitute the known values to the general rule`\text(Number of terms)``[n]` `=` `n` `\text(First term)``[a]` `=` `87` `\text(Common Difference)``[d]` `=` `-4` $$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{87}+[(\color{#9a00c7}{n}-1)\color{#00880A}{-4}]$$ Substitute known values `=` `87-4n+4` Distribute `=` `91-4n` `(iii)` Finding the `38th` termUse the answer from part `ii` to find the `38th` term$$U_{\color{#9a00c7}{n}}$$ `=` `91-4n` $$U_{\color{#9a00c7}{38}}$$ `=` `91-4(``38``)` Substitute known values `=` `91-152` Evaluate `=` `-61` `(iv)` Finding if `-132` is part of the sequenceUse the answer from part `ii` and have `U_n=-132`. Then solve for `n``U_n` `=` `91-4n` `-132` `=` `91-4n` Substitute `U_n=-132` `-132` `-91` `=` `91-4n` `-91` Subtract `91` from both sides `-223` `divide(-4)` `=` `-4n` `divide(-4)` Divide both sides by `-4` `55.75` `=` `n` `n` `=` `55.75` The value of `n` is not a whole integer. Therefore, `-132` is not part of the sequence`(i) \text(Yes)``(ii) U_n=91-4n``(iii) U_38=-61``(iv) \text(No)` -
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Question 3 of 6
3. Question
Given the expression `U_n=5n+2`, find:`(i)` The common difference`(ii)` The `10th` term-
`(i)` `d=` (5)`(ii)` `U_10=` (52)
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Help VideoCommon Difference Formula
$$d=U_2-U_1=U_3-U_2$$`(i)` Finding the common differenceFirst, substitute consecutive values to `n``U_n` `=` `5n+2` $$U_\color{#CC0000}{1}$$ `=` `5(``1``)+2` `=` `7` $$U_\color{#CC0000}{2}$$ `=` `5(``2``)+2` `=` `12` $$U_\color{#CC0000}{3}$$ `=` `5(``3``)+2` `=` `17` $$U_\color{#CC0000}{4}$$ `=` `5(``4``)+2` `=` `22` Next, use the formula to solve for the common difference`d` `=` `U_2-U_1` `=` `12-7` Substitute known values `=` `5` `(ii)` Finding the `10th` termSimply substitute `10` to `n``U_n` `=` `5n+2` $$U_\color{#CC0000}{10}$$ `=` `5(``10``)+2` Substitute `n=10` `=` `52` `(i) d=5``(ii) U_10=52` -
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Question 4 of 6
4. Question
Find the `18th` term`92,77,62,47…`- `U_18=` (-163)
Hint
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Help VideoCommon Difference Formula
$$d=U_2-U_1=U_3-U_2$$General Rule of an Arithmetic Sequence
$$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$First, solve for the value of `d`.$$\color{#00880A}{d}$$ `=` $$U_2-U_1$$ `=` `77-92` Substitute the first and second term `=` `-15` Next, substitute the known values to the general rule`\text(Number of terms)``[n]` `=` `18` `\text(First term)``[a]` `=` `92` `\text(Common Difference)``[d]` `=` `-15` $$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$U_{\color{#9a00c7}{18}}$$ `=` $$\color{#e65021}{92}+[(\color{#9a00c7}{18}-1)\color{#00880A}{-15}]$$ Substitute known values `=` `92+[17*(-15)]` Evaluate `=` `92-255` `=` `-163` `U_18=-163` -
Question 5 of 6
5. Question
Find the first positive term`1047-1012-977…`- `\text(First positive term) =` (3)
Hint
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Help VideoCommon Difference Formula
$$d=U_2-U_1=U_3-U_2$$General Rule of an Arithmetic Sequence
$$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$First, solve for the value of `d`.$$\color{#00880A}{d}$$ `=` $$U_2-U_1$$ `=` `-1012-(-1047)` Substitute the first and second term `=` `35` Next, transform the sequence into general rule form`\text(Number of terms)``[n]` `=` `n` `\text(First term)``[a]` `=` `-1047` `\text(Common Difference)``[d]` `=` `35` $$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{-1047}+[(\color{#9a00c7}{n}-1)\color{#00880A}{35}]$$ Substitute known values `=` `-1047+35n-35` Distribute `=` `35n-1082` Next, use the general rule form and solve for the value of `n` that is greater than `0``35n-1082` $$>$$ `0` `35n-1082` `+1082` $$>$$ `0` `+1082` Add `1082` to both sides `35n` `divide35` $$>$$ `1082` `divide35` Divide both sides by `35` `n` `=` `31` Rounded to a whole number Therefore, the first positive term would be the `31st` termFinally, use the general rule form to find the value of the `31st` term$$U_{\color{#9a00c7}{31}}$$ `=` $$35(\color{#9a00c7}{31})\color{#e65021}{-1082}$$ Substitute known values `=` `1085-1082` Evaluate `=` `3` `U_31=3` -
Question 6 of 6
6. Question
Find the `10th` term given that:`U_3=16``U_12=61`- `U_10=` (51)
Hint
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Help VideoGeneral Rule of an Arithmetic Sequence
$$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$First, transform the `3rd` and `12th` terms into general rule form3rd Term$$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$U_{\color{#9a00c7}{3}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{3}-1)\color{#00880A}{d}]$$ Substitute known values `16` `=` `a+2d` Distribute 12th Term$$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$U_{\color{#9a00c7}{12}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{12}-1)\color{#00880A}{d}]$$ Substitute known values `61` `=` `a+11d` Distribute Next, solve for the value of `d` by subtracting the `3rd` term’s general rule form from the `12th` term’s general rule form`61``-``16` `=` `(``a+2d``)-(``a+11d``)` `45``divide9` `=` `9d``divide9` Divide both sides by `9` `5` `=` `d` `d` `=` `5` Next, substitute `d` to one of the general rule forms to solve for `a``16` `=` `a+2``d` `16` `=` `a+2(``5``)` Substitute `d=5` `16` `-10` `=` `a+10` `-10` Subtract `10` from both sides `6` `=` `a` `a` `=` `6` Finally, substitute the known values to the general rule to find the `10th` term`\text(Number of terms)``[n]` `=` `10` `\text(First term)``[a]` `=` `6` `\text(Common Difference)``[d]` `=` `5` $$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$U_{\color{#9a00c7}{10}}$$ `=` $$\color{#e65021}{6}+[(\color{#9a00c7}{10}-1)\color{#00880A}{5}]$$ Substitute known values `=` `6+(9*5)` Evaluate `=` `6+45` `=` `51` `U_10=51`