Arithmetic sequences
What is the 15th term of the arithmetic sequence?
When the first term is 4 with a common difference of 6.
4, 7, 11, 15,...
What is the 15th term of the arithmetic sequence?
When the first term is 4 with a common difference of 6.
- Identify: a= 4 , d= 6
- Sub in term position, a, and b
- Solve
a15 = 4 + (15-1)6
a15 = 4 + (14)6
a15 = 88
Jackson and Marie want to purchase water bottles and socks for their school’s track team. The water bottles x, cost $12 each. While the socks y, cost $3 each. Jackson wants to spend less than $200, while Marie wants to spend no more than $100. Create a system of inequalities to model the situation.
The correct system of inequalities to model the situation is
12x + 3y < 200
12x + 3y < 100 .
Identify: Cost of each water bottle - $12
Cost of each pair of socks - $3
Jackson's budget - < 200
Marie's budget - < 100
Find the vertex of the following quadratic function. Then convert the function to vertex form f(x)= a(x-h)2 +k.
f(x)= x2 + 10x +12
The vertex form of the given quadratic function is f(x)= (x+5)2 -13.
- Use a and b to find the x - coordinate.
a=1 , b=10
h = -10/2(1)
= -5
- Substitute x= -5
f(-5)= -13
The vertex is (-5,-13)
Use the vertex form f(x)= a(x-h)2 +k and sub a=1, h= -5, and k= -13.
f(x)= 1(x -(-5) )2+(-13)
f(x)= (x+5)2 -13
Solve the absolute value inequality below.
|x+4|>5
The solution to the absolute value inequality |x+4|>5 is x< -9 or x> 1.
- Split into two cases, then solve
x+4>5 x+4<-5
-4 -4 -4 -4
x> 1 x< -9
Write a vertical motion model in the form 𝙝(𝙩)= -16t2 + v0t + h0. When the initial velocity is 32ft/s and the initial height is 20ft.
The vertical motion model to this situation is h(t) = -16t2 + 32t + 20.
The vertical motion model to symbolize this situation is h(t) = -16t2 + 32t + 20.
- Follow the vertical motion formula h(t) = -16t2 + v0t + h0 .
- Substitute the initial velocity v0t for 32.
- Substitute the initial height h0 for 20.
h(t)= -16t2 + 32t + 20
Find the 30th term of the arithmetic sequence. With a first term of 3 and a common difference of 5.
3, 8, 13, 18, 23...
The 30th term of the arithmetic sequence is 153.
- Identify: a= 3, d= 5
- Sub in term position, a, and b
- Solve
a30 = 3+(30 - 1)5
a30 = 3+(29)5
a30 = 148
A student wants to buy both apples and pears. They have a budget of $30 and know that apples (x) cost $3 each and pears (y) cost $1 each. They also need at least 13 fruits in total. Create a system of inequalities to model the budget and fruit requirement?
The system of inequalities to model the situation is 3x + 1y ≤ 30
x + y ≥ 13 .
Identify: Cost of each apple - $3
Cost of each pear - $ 1
Budget - ≤ 30
Number of fruits needed - ≥ 13
Find the vertex of the following quadratic function. Then convert the function to vertex form f(x)= a(x-h)2 +k.
f(x)= x2 +20x +30
The vertex form is f(x)= (x+10)2 -70
The vertex of the following function is (-10,-70).
- Use a and b to find the x - coordinate.
a= 1 , b= 20
h= -20/2(1)
h= -10
- Sub -10 for x
f(-10)= (-10)2 + 20(-10) +30
f(-10)= 100 - 200 + 30
f(-10)= -70
The vertex is (-10,-70).
- Use the vertex form f(x)= a(x-h)2+k and sub a= 1 , h= -10, and k= -70.
f(x)= 1(x-(-10) )2 + (-70)
f(x)= (x+10)2 -70
Solve the absolute value inequality below.
|x+6|>12
The solution to the absolute value inequality |x+6|>12 is x > 6 or x < -18.
- Split into two cases, then solve
|x+6|>12 |x+6|<-12
-6 -6 -6 -6
x > 6. x < -18
Write a vertical motion model in the form 𝙝(𝙩)= -16t2 + v0t + h0. When the initial velocity is 12ft/s and the initial height is 56ft.
The vertical motion model to represent this situation is h(t) = -16t2 + 12t + 56.
The vertical motion model to symbolize this situation is h(t) = -16t2 + 12t + 56.
- Follow the vertical motion formula h(t) = -16t2 + v0t + h0 .
- Substitute the initial velocity v0t for 12.
- Substitute the initial height h0 for 56
h(t)= -16t2 + 12t +56
The first term of an arithmetic sequence is 8 with a common difference of 13. What is the 60th term of the sequence?
The 60th term of the arithmetic sequence is 241.
- Identify: a= 8 , d= 13
- Sub in term position, a, and b
- Solve
a60 = 8+(60-1)13
a60 = 8+(59)13
a60 = 775
Mariyah is selling bracelets and earrings to make money for a vacation. Bracelets (Y) cost $3 each, and earrings (X) cost $5 each. She needs to make at least $500, and she knows she will sell more than 50 bracelets. Create a system of inequalities to represent the situation.
The system of inequalities to represent the situation is
5x + 3y > 50
x + y ≥ 500 .
Identify: Cost of each bracelet - $3
Cost of each earring - $ 5
Budget - ≥ 500
Number of bracelets that have to be sold - >50
Find the vertex of the following quadratic function. Then convert the function to vertex form f(x)= a(x-h)2 +k.
f(x)= x2+30x+70
The vertex form of the following quadratic equation is f(x)= 1(x+15)2 -155.
The vertex of the following function is (15,-155).
- Use a and b to find the x - coordinate.
a= 1 , b= 30
h= -30/2(1)
h= -15
- Sub -15 for x
f(-15)= (-15)2 + 30(-15) +30
f(-15)= 225 - 450 +30
f(-15)= -155
The vertex is (-15,-155).
- Use the vertex form f(x)= a(x-h)2+k and sub a= 1 , h= -15, and k= -155.
f(x)= 1(x-(-15) )2 + (-155)
f(x)= (x+15)2 -155
Solve the absolute value inequality below.
|x+2|<5
The solution to the inequality |x+2|<8 is -7 < x < 3.
Using the property of absolute value inequalities. If |x|<a, then −a <x< a
|x+2|<5 implies -5 <x+2< 5
- Solve for x by subtracting 2 from all parts of the inequality.
-5 < x + 2 < 5
-2 -2 -2
-7 < x < 5
Write a vertical motion model in the form 𝙝(𝙩)= -16t2 + v0t + h0. When the initial velocity is 60ft/s and the initial height is 323ft
The vertical motion model to symbolize this situation is h(t) = -16t2 + 60t + 323.
- Follow the vertical motion formula h(t) = -16t2 + v0t + h0 .
- Substitute the initial velocity v0t for 60.
- Substitute the initial height h0 for 323.
h(t)= -16t + 60t +323