Arithmetic Sequences

Question 1 of 6

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

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

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

Find the

$18th$ term

$92,77,62,47…$

$U_18=$ (-163)

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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

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

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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Common Difference Formula

General Rule of an Arithmetic Sequence

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General Rule of an Arithmetic Sequence

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General Rule of an Arithmetic Sequence

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