Question

topic 5: arithmetic and geometric sequences

1. determine if the sequence is arithmetic or geometric. then write the explicit rule.

50, 40, 30, 20, ...

1. determine if the sequence is arithmetic or geometric. then write the explicit rule.

50, 25, 12.5, 6.25, ...

1. write the first 4 terms of this arithmetic sequence: ( a(n) = 42 + 3(n - 1) )
2. write the first 4 terms of this geometric sequence: ( g(n) = 12(3)^{n - 1} )
3. what is the value of the 100th term?( a(n) = 20 - 4(n - 1) )
4. which term has a value of 196?( a(n) = 8 + 5(n - 1) )

topic 6: slope and intercepts

1. find the slope: ( (5, 2) ) ( (10, 22) )
2. find the slope: ( (-7, 8) ) ( (-2, 3) )
3. write the slope - intercept form:( (0, 6) ) ( (12, 18) )
4. write the slope - intercept form:( (-1, 3) ) ( (5, 2) )
5. write the equation of each of these lines:

(image of two intersecting lines on a coordinate plane)
topic 7: point - slope and standard form

1. write the point - slope form. then convert it into standard form.( (10, 20) ) ( (30, -40) )
2. rewrite this standard form equation into slope - intercept form.( 4x - 5y = 40 )
3. rewrite this slope - intercept form in standard form.( y=\frac{2}{3}x + 8 )
4. find the intercepts of this equation. make an intercept table.( -3x + 12y = 36 )

Explanation:

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Question 25

Step1: Identify sequence type

Check differences: $40-50=-10$, $30-40=-10$, $20-30=-10$ (constant difference, arithmetic)

Step2: Find explicit rule

Arithmetic rule: $a(n)=a_1+(n-1)d$, $a_1=50$, $d=-10$
$a(n)=50+(n-1)(-10)=60-10n$

Question 26

Step1: Identify sequence type

Check ratios: $\frac{25}{50}=0.5$, $\frac{12.5}{25}=0.5$, $\frac{6.25}{12.5}=0.5$ (constant ratio, geometric)

Step2: Find explicit rule

Geometric rule: $g(n)=g_1r^{n-1}$, $g_1=50$, $r=0.5$
$g(n)=50(0.5)^{n-1}$

Question 27

Step1: Calculate term 1

Substitute $n=1$: $a(1)=42+3(1-1)=42$

Step2: Calculate term 2

Substitute $n=2$: $a(2)=42+3(2-1)=45$

Step3: Calculate term 3

Substitute $n=3$: $a(3)=42+3(3-1)=48$

Step4: Calculate term 4

Substitute $n=4$: $a(4)=42+3(4-1)=51$

Question 28

Step1: Calculate term 1

Substitute $n=1$: $g(1)=12(3)^{1-1}=12$

Step2: Calculate term 2

Substitute $n=2$: $g(2)=12(3)^{2-1}=36$

Step3: Calculate term 3

Substitute $n=3$: $g(3)=12(3)^{3-1}=108$

Step4: Calculate term 4

Substitute $n=4$: $g(4)=12(3)^{4-1}=324$

Question 29

Step1: Substitute $n=100$

$a(100)=20-4(100-1)$

Step2: Simplify expression

$a(100)=20-4(99)=20-396=-376$

Question 30

Step1: Set $a(n)=196$, solve for $n$

$196=8+5(n-1)$

Step2: Isolate the variable term

$196-8=5(n-1) \implies 188=5(n-1)$

Step3: Solve for $n$

$n-1=\frac{188}{5}=37.6 \implies n=38.6$ (correction: $196-8=188$ → $188\div5=37.6$, but since $n$ must be integer, recheck: $8+5(n-1)=196$ → $5(n-1)=188$ → no integer solution? Wait, $8+5(38-1)=8+185=193$, $8+5(39-1)=8+190=198$. *Note: If we assume no typo, $n=38.6$, but standard sequences use integer $n$.*

Question 31

Step1: Use slope formula $m=\frac{y_2-y_1}{x_2-x_1}$

$(x_1,y_1)=(5,2)$, $(x_2,y_2)=(10,22)$

Step2: Calculate slope

$m=\frac{22-2}{10-5}=\frac{20}{5}=4$

Question 32

Step1: Use slope formula

$(x_1,y_1)=(-7,8)$, $(x_2,y_2)=(-2,3)$

Step2: Calculate slope

$m=\frac{3-8}{-2-(-7)}=\frac{-5}{5}=-1$

Question 33

Step1: Calculate slope

$(x_1,y_1)=(0,6)$, $(x_2,y_2)=(12,18)$
$m=\frac{18-6}{12-0}=\frac{12}{12}=1$

Step2: Write slope-intercept form ($y=mx+b$)

$b=6$ (y-intercept, from $(0,6)$), so $y=x+6$

Question 34

Step1: Calculate slope

$(x_1,y_1)=(-1,3)$, $(x_2,y_2)=(5,2)$
$m=\frac{2-3}{5-(-1)}=\frac{-1}{6}=-\frac{1}{6}$

Step2: Find $b$ using $y=mx+b$

Substitute $(-1,3)$: $3=-\frac{1}{6}(-1)+b \implies 3=\frac{1}{6}+b \implies b=\frac{17}{6}$

Step3: Write final equation

$y=-\frac{1}{6}x+\frac{17}{6}$

Question 35

Line 1 (positive slope):

Step1: Identify intercepts

Y-intercept $b=1$, x-intercept $(-1,0)$

Step2: Calculate slope

$m=\frac{1-0}{0-(-1)}=1$

Step3: Write equation

$y=x+1$

Line 2 (negative slope):

Step1: Identify intercepts

Y-intercept $b=-3$, x-intercept $(-3,0)$

Step2: Calculate slope

$m=\frac{-3-0}{0-(-3)}=-1$

Step3: Write equation

$y=-x-3$

Question 37

Step1: Calculate slope

$(x_1,y_1)=(10,20)$, $(x_2,y_2)=(30,-40)$
$m=\frac{-40-20}{30-10}=\frac{-60}{20}=-3$

Step2: Write point-slope form

Use $(10,20)$: $y-20=-3(x-10)$

Step3: Convert to standard form ($Ax+By=C$)

$y-20=-3x+30 \implies 3x+y=50$

Question 38

Step1: Isolate $y$ to solve

$4x-5y=40$

Step2: Rearrange into slope-intercept form

$-5y=-4x+40 \implies y=\frac{4}{5}x-8$

Question 39

Step1: Eliminate fraction, rearrange

$y=\frac{2}{3}x+8$ → multiply by 3: $3y=2x+24$

Step2: Write standard form

$-2x+3y=24$ or $2x-3y=-24$

Question 40

Step1: Find x-intercept (set $y=0$)

$-3x+12(0)=36 \implies -3x=36 \implies x=-12$

Step2: Find y-intercept (set $x=0$)

$-3(0)+12y=36 \implies 12y=36 \implies y=3$

Step3: Create intercept table

| Intercept Type | Value | Coordinate |
|----------------|-------|------------|
| X-intercept | $-12$ | $(-12, 0)$ |
| Y-intercept | $3$ | $(0, 3)$ |

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Answer:

1. Arithmetic sequence; $a(n)=60-10n$
2. Geometric sequence; $g(n)=50(0.5)^{n-1}$
3. 42, 45, 48, 51
4. 12, 36, 108, 324
5. $-376$
6. No integer term equals 196; $n=38.6$
7. $4$
8. $-1$
9. $y=x+6$
10. $y=-\frac{1}{6}x+\frac{17}{6}$
11. Positive slope line: $y=x+1$; Negative slope line: $y=-x-3$
12. Point-slope: $y-20=-3(x-10)$; Standard: $3x+y=50$
13. $y=\frac{4}{5}x-8$
14. $2x-3y=-24$

40)

Intercept Type	Value	Coordinate
Y-intercept	$3$	$(0, 3)$