Arithmetic Sequences

Question 1 of 6

Given the sequence

1 + 5 + 9 \dots

$1+5+9…$ , find:

(i)

$(i)$ The common difference

(i i)

$(ii)$ The

11 t h

$11th$ term

$(i)$ $(i)$ $d =$ $d=$ (4)

$(i i)$ $(ii)$ $U_{11} =$ $U_11=$ (41)

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Common Difference Formula

d = U_{2} - U_{1} = U_{3} - U_{2}

$d=U_2-U_1=U_3-U_2$

General Rule of an Arithmetic Sequence

U_{n} = a + [(n - 1) d]

$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$

(i)

$(i)$ Finding the common difference

First, identify the consecutive values

$U_\color{#CC0000}{1}$	$=$ $=$	$1$ $1$
$U_\color{#CC0000}{2}$	$=$ $=$	$5$ $5$
$U_\color{#CC0000}{3}$	$=$ $=$	$9$ $9$

Next, use the formula to solve for the common difference

$d$ $d$	$=$ $=$	$U_{2} - U_{1}$ $U_2-U_1$
	$=$ $=$	$5 - 1$ $5-1$	Substitute known values
	$=$ $=$	$4$ $4$

(i i)

$(ii)$ Finding the

10 t h

$10th$ term

Substitute the known values to the general rule

$Number of terms$ $\text(Number of terms)$ $[n]$ $[n]$	$=$ $=$	$11$ $11$
$First term$ $\text(First term)$ $[a]$ $[a]$	$=$ $=$	$1$ $1$
$Common Difference$ $\text(Common Difference)$ $[d]$ $[d]$	$=$ $=$	$4$ $4$

$U_{\color{#9a00c7}{n}}$	$=$ $=$	$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$
$U_{\color{#9a00c7}{11}}$	$=$ $=$	$\color{#e65021}{1}+[(\color{#9a00c7}{11}-1)\color{#00880A}{4}]$	Substitute known values
	$=$ $=$	$1 + [(10) 4]$ $1+[(10)4]$	Evaluate
	$=$ $=$	$41$ $41$

(i) d = 4

$(i) d=4$

(i i) U_{11} = 41

$(ii) U_11=41$

Question 2 of 6

Given the sequence

87, 83, 79, 75 \dots

$87,83,79,75…$ , find:

(i)

$(i)$ If the expression is an arithmetic sequence

(i i)

$(ii)$ The general rule form

(i i i)

$(iii)$ The

38 t h

$38th$ term

(i v)

$(iv)$ If

- 132

$-132$ is part of the sequence

For part

i

$i$ and

i v

$iv$ , write

Y

$Y$ for yes and

N

$N$ for no

$(i)$ $(i)$ (Y, y)

$(i i)$ $(ii)$ $U_{n} =$ $U_n=$ (91-4n)

$(i i i)$ $(iii)$ $U_{38} =$ $U_38=$ (-61)

$(i v)$ $(iv)$ (N, n)

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Common Difference Formula

d = U_{2} - U_{1} = U_{3} - U_{2}

$d=U_2-U_1=U_3-U_2$

General Rule of an Arithmetic Sequence

U_{n} = a + [(n - 1) d]

$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$

(i)

$(i)$ Finding if the expression is an arithmetic sequence

First, identify the consecutive values

$U_\color{#CC0000}{1}$	$=$ $=$	$1$ $1$
$U_\color{#CC0000}{2}$	$=$ $=$	$5$ $5$
$U_\color{#CC0000}{3}$	$=$ $=$	$9$ $9$

Next, use the formula to solve for the common difference and check if it is consistent with the consecutive values

$d$ $d$	$=$ $=$	$U_{2} - U_{1}$ $U_2-U_1$
	$=$ $=$	$83 - 87$ $83-87$	Substitute known values
	$=$ $=$	$- 4$ $-4$

$d$ $d$	$=$ $=$	$U_{3} - U_{2}$ $U_3-U_2$
	$=$ $=$	$79 - 83$ $79-83$	Substitute known values
	$=$ $=$	$- 4$ $-4$

$d$ $d$	$=$ $=$	$U_{4} - U_{3}$ $U_4-U_3$
	$=$ $=$	$75 - 79$ $75-79$	Substitute known values
	$=$ $=$	$- 4$ $-4$

The common difference is consistent. Therefore, the expression is an arithmetic sequence.

(i i)

$(ii)$ Finding the general rule form

Substitute the known values to the general rule

$Number of terms$ $\text(Number of terms)$ $[n]$ $[n]$	$=$ $=$	$n$ $n$
$First term$ $\text(First term)$ $[a]$ $[a]$	$=$ $=$	$87$ $87$
$Common Difference$ $\text(Common Difference)$ $[d]$ $[d]$	$=$ $=$	$- 4$ $-4$

$U_{\color{#9a00c7}{n}}$	$=$ $=$	$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$
$U_{\color{#9a00c7}{n}}$	$=$ $=$	$\color{#e65021}{87}+[(\color{#9a00c7}{n}-1)\color{#00880A}{-4}]$	Substitute known values
	$=$ $=$	$87 - 4 n + 4$ $87-4n+4$	Distribute
	$=$ $=$	$91 - 4 n$ $91-4n$

(i i i)

$(iii)$ Finding the

38 t h

$38th$ term

Use the answer from part

i i

$ii$ to find the

38 t h

$38th$ term

$U_{\color{#9a00c7}{n}}$	$=$ $=$	$91 - 4 n$ $91-4n$
$U_{\color{#9a00c7}{38}}$	$=$ $=$	$91 - 4 ($ $91-4($ $38$ $38$ $)$ $)$	Substitute known values
	$=$ $=$	$91 - 152$ $91-152$	Evaluate
	$=$ $=$	$- 61$ $-61$

(i v)

$(iv)$ Finding if

- 132

$-132$ is part of the sequence

Use the answer from part

i i

$ii$ and have

U_{n} = - 132

$U_n=-132$ . Then solve for

n

$n$

$U_{n}$ $U_n$	$=$ $=$	$91 - 4 n$ $91-4n$
$- 132$ $-132$	$=$ $=$	$91 - 4 n$ $91-4n$	Substitute $U_{n} = - 132$ $U_n=-132$
$- 132$ $-132$ $- 91$ $-91$	$=$ $=$	$91 - 4 n$ $91-4n$ $- 91$ $-91$	Subtract $91$ $91$ from both sides
$- 223$ $-223$ $\div (- 4)$ $divide(-4)$	$=$ $=$	$- 4 n$ $-4n$ $\div (- 4)$ $divide(-4)$	Divide both sides by $- 4$ $-4$
$55.75$ $55.75$	$=$ $=$	$n$ $n$
$n$ $n$	$=$ $=$	$55.75$ $55.75$

The value of

n

$n$ is not a whole integer. Therefore,

- 132

$-132$ is not part of the sequence

(i) Yes

$(i) \text(Yes)$

(i i) U_{n} = 91 - 4 n

$(ii) U_n=91-4n$

(i i i) U_{38} = - 61

$(iii) U_38=-61$

(i v) No

$(iv) \text(No)$

Question 3 of 6

Given the expression

U_{n} = 5 n + 2

$U_n=5n+2$ , find:

(i)

$(i)$ The common difference

(i i)

$(ii)$ The

10 t h

$10th$ term

$(i)$ $(i)$ $d =$ $d=$ (5)

$(i i)$ $(ii)$ $U_{10} =$ $U_10=$ (52)

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Common Difference Formula

d = U_{2} - U_{1} = U_{3} - U_{2}

$d=U_2-U_1=U_3-U_2$

(i)

$(i)$ Finding the common difference

First, substitute consecutive values to

n

$n$

$U_{n}$ $U_n$	$=$ $=$	$5 n + 2$ $5n+2$
$U_\color{#CC0000}{1}$	$=$ $=$	$5 ($ $5($ $1$ $1$ $) + 2$ $)+2$	$=$ $=$	$7$ $7$
$U_\color{#CC0000}{2}$	$=$ $=$	$5 ($ $5($ $2$ $2$ $) + 2$ $)+2$	$=$ $=$	$12$ $12$
$U_\color{#CC0000}{3}$	$=$ $=$	$5 ($ $5($ $3$ $3$ $) + 2$ $)+2$	$=$ $=$	$17$ $17$
$U_\color{#CC0000}{4}$	$=$ $=$	$5 ($ $5($ $4$ $4$ $) + 2$ $)+2$	$=$ $=$	$22$ $22$

Next, use the formula to solve for the common difference

$d$ $d$	$=$ $=$	$U_{2} - U_{1}$ $U_2-U_1$
	$=$ $=$	$12 - 7$ $12-7$	Substitute known values
	$=$ $=$	$5$ $5$

(i i)

$(ii)$ Finding the

10 t h

$10th$ term

Simply substitute

10

$10$ to

n

$n$

$U_{n}$ $U_n$	$=$ $=$	$5 n + 2$ $5n+2$
$U_\color{#CC0000}{10}$	$=$ $=$	$5 ($ $5($ $10$ $10$ $) + 2$ $)+2$	Substitute $n = 10$ $n=10$
	$=$ $=$	$52$ $52$

(i) d = 5

$(i) d=5$

(i i) U_{10} = 52

$(ii) U_10=52$

Question 4 of 6

Find the

18 t h

$18th$ term

92, 77, 62, 47 \dots

$92,77,62,47…$

$U_{18} =$ $U_18=$ (-163)

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Common Difference Formula

d = U_{2} - U_{1} = U_{3} - U_{2}

$d=U_2-U_1=U_3-U_2$

General Rule of an Arithmetic Sequence

U_{n} = a + [(n - 1) d]

$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$

First, solve for the value of $d$ $d$ .

$\color{#00880A}{d}$	$=$ $=$	$U_2-U_1$

	$=$ $=$	$77 - 92$ $77-92$	Substitute the first and second term

	$=$ $=$	$- 15$ $-15$

Next, substitute the known values to the general rule

$Number of terms$ $\text(Number of terms)$ $[n]$ $[n]$	$=$ $=$	$18$ $18$
$First term$ $\text(First term)$ $[a]$ $[a]$	$=$ $=$	$92$ $92$
$Common Difference$ $\text(Common Difference)$ $[d]$ $[d]$	$=$ $=$	$- 15$ $-15$

$U_{\color{#9a00c7}{n}}$	$=$ $=$	$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$
$U_{\color{#9a00c7}{18}}$	$=$ $=$	$\color{#e65021}{92}+[(\color{#9a00c7}{18}-1)\color{#00880A}{-15}]$	Substitute known values
	$=$ $=$	$92 + [17 \cdot (- 15)]$ $92+[17*(-15)]$	Evaluate
	$=$ $=$	$92 - 255$ $92-255$
	$=$ $=$	$- 163$ $-163$

U_{18} = - 163

$U_18=-163$

Question 5 of 6

Find the first positive term

1047 - 1012 - 977 \dots

$1047-1012-977…$

$First positive term =$ $\text(First positive term) =$ (3)

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Common Difference Formula

d = U_{2} - U_{1} = U_{3} - U_{2}

$d=U_2-U_1=U_3-U_2$

General Rule of an Arithmetic Sequence

U_{n} = a + [(n - 1) d]

$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$

First, solve for the value of $d$ $d$ .

$\color{#00880A}{d}$	$=$ $=$	$U_2-U_1$

	$=$ $=$	$- 1012 - (- 1047)$ $-1012-(-1047)$	Substitute the first and second term

	$=$ $=$	$35$ $35$

Next, transform the sequence into general rule form

$Number of terms$ $\text(Number of terms)$ $[n]$ $[n]$	$=$ $=$	$n$ $n$
$First term$ $\text(First term)$ $[a]$ $[a]$	$=$ $=$	$- 1047$ $-1047$
$Common Difference$ $\text(Common Difference)$ $[d]$ $[d]$	$=$ $=$	$35$ $35$

$U_{\color{#9a00c7}{n}}$	$=$ $=$	$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$
$U_{\color{#9a00c7}{n}}$	$=$ $=$	$\color{#e65021}{-1047}+[(\color{#9a00c7}{n}-1)\color{#00880A}{35}]$	Substitute known values
	$=$ $=$	$- 1047 + 35 n - 35$ $-1047+35n-35$	Distribute
	$=$ $=$	$35 n - 1082$ $35n-1082$

Next, use the general rule form and solve for the value of $n$ $n$ that is greater than

0

$0$

$35 n - 1082$ $35n-1082$	$>$	$0$ $0$
$35 n - 1082$ $35n-1082$ $+ 1082$ $+1082$	$>$	$0$ $0$ $+ 1082$ $+1082$	Add $1082$ $1082$ to both sides
$35 n$ $35n$ $\div 35$ $divide35$	$>$	$1082$ $1082$ $\div 35$ $divide35$	Divide both sides by $35$ $35$
$n$ $n$	$=$ $=$	$31$ $31$	Rounded to a whole number

Therefore, the first positive term would be the

31 s t

$31st$ term

Finally, use the general rule form to find the value of the

31 s t

$31st$ term

$U_{\color{#9a00c7}{31}}$	$=$ $=$	$35(\color{#9a00c7}{31})\color{#e65021}{-1082}$	Substitute known values
	$=$ $=$	$1085 - 1082$ $1085-1082$	Evaluate
	$=$ $=$	$3$ $3$

U_{31} = 3

$U_31=3$

Question 6 of 6

Find the

10 t h

$10th$ term given that:

U_{3} = 16

$U_3=16$

U_{12} = 61

$U_12=61$

$U_{10} =$ $U_10=$ (51)

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General Rule of an Arithmetic Sequence

U_{n} = a + [(n - 1) d]

$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$

First, transform the

3 r d

$3rd$ and

12 t h

$12th$ 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
$16$ $16$	$=$ $=$	$a + 2 d$ $a+2d$	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
$61$ $61$	$=$ $=$	$a + 11 d$ $a+11d$	Distribute

Next, solve for the value of $d$ $d$ by subtracting the $3 r d$ $3rd$ term’s general rule form from the $12 t h$ $12th$ term’s general rule form

$61$ $61$ $-$ $-$ $16$ $16$	$=$ $=$	$($ $($ $a + 2 d$ $a+2d$ $) - ($ $)-($ $a + 11 d$ $a+11d$ $)$ $)$
$45$ $45$ $\div 9$ $divide9$	$=$ $=$	$9 d$ $9d$ $\div 9$ $divide9$	Divide both sides by $9$ $9$
$5$ $5$	$=$ $=$	$d$ $d$
$d$ $d$	$=$ $=$	$5$ $5$

Next, substitute $d$ $d$ to one of the general rule forms to solve for $a$ $a$

$16$ $16$	$=$ $=$	$a + 2$ $a+2$ $d$ $d$
$16$ $16$	$=$ $=$	$a + 2 ($ $a+2($ $5$ $5$ $)$ $)$	Substitute $d = 5$ $d=5$
$16$ $16$ $- 10$ $-10$	$=$ $=$	$a + 10$ $a+10$ $- 10$ $-10$	Subtract $10$ $10$ from both sides
$6$ $6$	$=$ $=$	$a$ $a$
$a$ $a$	$=$ $=$	$6$ $6$

Finally, substitute the known values to the general rule to find the

10 t h

$10th$ term

$Number of terms$ $\text(Number of terms)$ $[n]$ $[n]$	$=$ $=$	$10$ $10$
$First term$ $\text(First term)$ $[a]$ $[a]$	$=$ $=$	$6$ $6$
$Common Difference$ $\text(Common Difference)$ $[d]$ $[d]$	$=$ $=$	$5$ $5$

$U_{\color{#9a00c7}{n}}$	$=$ $=$	$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$
$U_{\color{#9a00c7}{10}}$	$=$ $=$	$\color{#e65021}{6}+[(\color{#9a00c7}{10}-1)\color{#00880A}{5}]$	Substitute known values
	$=$ $=$	$6 + (9 \cdot 5)$ $6+(9*5)$	Evaluate
	$=$ $=$	$6 + 45$ $6+45$
	$=$ $=$	$51$ $51$

U_{10} = 51

$U_10=51$

Arithmetic Sequences

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1. Question

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Common Difference Formula

General Rule of an Arithmetic Sequence

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Common Difference Formula

General Rule of an Arithmetic Sequence

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Common Difference Formula

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General Rule of an Arithmetic Sequence

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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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