Intro to Sequences

Sequences and Series

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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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Common 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 difference

First, 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` term

Substitute 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`

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 sequence

For 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)

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Common 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 sequence

First, 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 form

Substitute 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` term

Use 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 sequence

Use 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)`

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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Common Difference Formula

$$d=U_2-U_1=U_3-U_2$$

`(i)` Finding the common difference

First, 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` term

Simply 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`

Question 4 of 6

4. Question

Find the `18th` term

`92,77,62,47…`

`U_18=` (-163)

Hint

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Common 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)

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Common 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` term

Finally, 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)

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General 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 form

3rd 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`

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Intro to Sequences
Arithmetic Series (Sum)
Geometric Series (Sum)
Infinite Geometric Series