Percent change of 88%
1) Consider 50 as equivalent to 100% as well as 94 to x.
50 ----------- 100%
94 ------------ x
2) Now we can set the ratios and cross multiply then:
[tex]\begin{gathered} \frac{50}{94}=\frac{100}{x} \\ 50x=94\cdot100 \\ 50x=9400 \\ \frac{50x}{50}=\frac{9400}{50} \\ x=188 \\ \end{gathered}[/tex]So 94 corresponds to 188%. Then we can see that there was an increase. Since 94 >50.
But we need to subtract from 100% to find out the percent change from 50 to 94.
188%-100%= 88%
3) Hence, the answer is an increase of 88%
On a piece of paper, graph yz 2x - 3. Then determine which answer choicematches the graph you drew.ABсD(2, 1)(2, 1)(2, 1)((2, 1)(0-3)0,-3)(0-3)(0-3)A. Graph DOB. Graph BO C. Graph AD. Graph
The graph says
[tex]y\ge x-1[/tex]The upper section of the graph will be shaded since y is greater than or equals to x - 1.
The answer is A . Graph A
For each expression, combine like terms and write an equivalent expression with fewer terms.a. 4x+3xb. 3x+5x-1c. 5+2x+7+4xd. 4-2x+5xe. 10x-5+3x-2
To simplify the expressions you have to combine the like terms.
This means that you'll solve the operations between the terms that have the same variables, for example x + 2x=3x
Or the terms that have no variables and are only numbers, for example 4+5=9
a. The expression is
[tex]4x+3x[/tex]Both terms have the same variable "x", so you can add them together. To do so, add the coefficients, i.e. the numbers that are being multiplied by x
[tex]4x+3x=(4+3)x=7x[/tex]And you get that the simplified expression is 7x
b. The expression is
[tex]3x+5x-1[/tex]In this expression you have two types of terms, the x-related terms and one constant. In this case you have to solve the operation for the x-related terms together and leave the constant as it is
[tex](3x+5x)-1=(3+5)x-1=8x-1[/tex]The simplified expression is 8x-1
Line a is parallel to line b and line c is parallel to line d, using the diagram what can be said about angle 7 and 12
Answer:
C
Step-by-step explanation:
Angle 7 is congruent to angle 5 by the corresponding angles theorem, and angles 5 and 12 are supplementary because they are consecutive interior angles.
Thus, angles 7 and 12 are supplementary.
Rick Takei has a 4-wheel drive vehicle whose average retail value is $15,857. A used vehicle guide adds $60 for heated outside mirrors, $250 for rear and side air bags. $175 for cruise control, and $100 for remote keyless entry. It suggests deducting $750 for excessive mileage. What is the average retail price?
We are asked to find the final average retail price of the vehicle after the given additions and deductions.
The average retail value is $15,857
Add $60 for heated outside mirrors.
[tex]$\$15,857+\$60=\$15,917$[/tex]Add $250 for rear and side airbags.
[tex]\$15,917+\$250=\$16,167[/tex]Add $175 for cruise control.
[tex]\$16,167+\$175=\$16,342[/tex]Add $100 for remote keyless entry.
[tex]\$16,342+\$100=\$16,442[/tex]Deduct $750 for excessive mileage.
[tex]\$16,442-\$750=\$15,692[/tex]Therefore, the average retail price is $15,692
Consider that AABC is similar to AXYZ and the measure of ZB is 68º. What is the measure of ZY? A) 70° B) 68° C) 41° D) 22°
Answer
Option B is correct.
Angle Y = 68º
Explanation
Similar triangles have the same set of angles in them.
All the corresponding angles are equal to each other.
So, if triangle ABC is similar to triangle XYZ
Angle A = Angle X
Angle B = Angle Y
Angle C = Angle Z
The order in which they are named determines the angles that are corresponding to each other.
So, if Angle B = 68º
Angle Y = Angle B = 68º
Hope this Helps!!!
Give three value to x such that |xl= -x.
We have the following equation:
[tex]\left|x\right|=-x[/tex]And we want to identify values who satisfy the equation. And the possible answers for this case are:
x=0
Since :
[tex]\left|0\right|=-0=0[/tex]Other two possible answers are:
[tex]\left|\frac{0}{10}\right|=-\frac{0}{10}=0[/tex]In the following figure PR is perpendicular to QS. PR= 14cm, QR=15 cm and QS =12 cm. What is the perimeter of PQR?
The width of a Rectangle is 3.6 inches and the perimeter is 72 inches. What is the length of the rectangle?
We know that
• The width of the rectangle is 3.6 inches.
,• The perimeter is 72 inches.
The perimeter formula for a rectangle is
[tex]P=2(w+l)[/tex]Where P = 72, w = 3.6, and we have to solve for l.
[tex]\begin{gathered} 72=2(3.6+l) \\ \frac{72}{2}=3.6+l \\ 36=3.6+l \\ l=36-3.6 \\ l=32.4 \end{gathered}[/tex]Therefore, the length of the rectangle is 32.4 inches.Is z(x) = 1 + 6x^2 + 4x a quadratic function? I don't believe it's linear or constant, just wanted to make sure.
Given function is
[tex]z(x)=1+6x^2+4x[/tex]In the given function, the maximum power of x is 2 and it is of the form
[tex]ax^2+bx+c[/tex]Therefore, it is a quadratic function.
Point m represents the opposite of negative 1/2 and point n represents the opposite of positive 5/2 which number line correctly shows points m and n great
If Point m represents the opposite of negative 1/2 and point n represents the opposite of positive 5/2. Then M is 1/2 and N is -5/2.
What is Number system?A number system is defined as the representation of numbers by using digits or other symbols in a consistent manner.
A number line is a picture of a graduated straight line that serves as visual representation of the real numbers.
Given that point M represents the opposite of negative 1/2. Which means opposite of -1/2. The opposite of -1/2 means positive of 1/2. The sign changes.
Point N represents the opposite of positive 5/2. Which means opposite of 5/2. The opposite of 5/2 means negative of 5/2. The sign changes.
opposite of positive 5/2 is -5/2.
Now let us plot this values on a number line. 1/2 is 0.5 and -5/2 means -2.5.
The graph is attached below.
Hence M is 1/2 and N is -5/2
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A toy costs 35 000 lndonesian rupiah (Rp).
The conversion rate is Rp 1000 = 5$0.145 598.
Without using a calculator, estimate the price of
the toy in S$.
here is your answer I hope this helps
3. Which of the following equations would be a parabola with vertex (2,-3) that opendownwards? Select ALL.a.h.y = -2(x - 2)2 – 3i.y = (-x + 2)2 + 3C.y = -(x - 2)2 – 3b. y = -(x + 2)2 – 3y = -(x - 2)2 + 3d. y=-(x + 2)2 +3y = -(x - 2)3 +3f. y = -(x - 2)3 – 3y=-{(x - 2)2 - 3j. y = (-x - 2)2 – 3k. y = (-x + 2)2 – 3: نه1. y = (-x - 2)2 – 3m. y =} (x - 2)2 – 3g.n.y = 2(x - 2)2 - 3
Solution:
Given:
[tex]\text{Parabola with vertex (2,-3) that open downwards}[/tex]The equation of a parabola in vertex form is given by;
[tex]\begin{gathered} y=a(x-h)^2+k \\ \\ \text{where;} \\ (h,k)\text{ is the vertex} \\ \\ \text{Hence,} \\ h=2 \\ k=-3 \end{gathered}[/tex]Hence, the equation of the parabola is;
[tex]\begin{gathered} y=a(x-2)^2-3 \\ \\ \text{For the parabola to open downwards, then;} \\ a<0 \\ a\text{ must be negative} \end{gathered}[/tex]Hence, from the options, the equations that have a as negative and in the form gotten above will be selected.
Therefore, the equations of a parabola with vertex (2,-3) that open downwards are;
[tex]\begin{gathered} y=-2(x-2)^2-3 \\ \\ y=-(x-2)^2-3 \end{gathered}[/tex]Bonus: Write the equation of a line in slope intercept form that is parallelto y=4/3x-7 and contains the point (5,-8)
The given equation is
[tex]y=\frac{4}{3}x-7[/tex]The slope of the given line is 4/3 because it's the coefficient of x.
Now, the new line has a slope of 4/3 too because parallel lines have equal slopes.
We know that the new line passes through the point (5, -8). Let's use the point-slope formula to find the equation.
[tex]y-y_1=m(x-x_1)[/tex]Replacing the points and the slope, we have.
[tex]\begin{gathered} y-(-8)=\frac{4}{3}(x-5) \\ y+8=\frac{4}{3}x-\frac{20}{3} \\ y=\frac{4}{3}x-\frac{20}{3}-8 \\ y=\frac{4}{3}x+\frac{-20-24}{3} \\ y=\frac{4}{3}x-\frac{44}{3} \end{gathered}[/tex]Therefore, the equation of the new line is y = (4/3)x - (44/3).A firefighter has an annual income of $46,870. The income tax the firefighter has to pay is 16%. What is the amount of income tax in dollars and cents that the firefighter has to pay? (TEKS 7.13A-S)
amount of income tax:
[tex]Tax=46,870\times0.16=7499.2[/tex]Answer:
$7499.2
the quotient of two numbers is -1 their difference is 8 what are the numbers
Let the two numbers be represented with x and y.
Quotient of two numbers = -1:
[tex]\frac{x}{y}=-1\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\text{.}.(1)[/tex]Difference of two numbers = 8:
[tex]x\text{ - y = 8}\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots(2)[/tex]From the first equation, make x the the subject.
Thus, we have:
x = -y
Substitute -y for x in equation 2:
-y - y = 8
-2y = 8
Divide both sides by -2:
[tex]\begin{gathered} \frac{-2y}{-2}=\frac{8}{-2} \\ \\ y\text{ = -4} \end{gathered}[/tex]Now, substitute -4 for y in equation 2:
x - y = 8
x - (-4) = 8
x + 4 = 8
Subtract 4 from both sides:
x + 4 - 4 = 8 - 4
x = 4
Therefore,
x = 4 and y = -4
Thus, the numbers are 4 and -4
ANSWER:
4 and -4
Find the area of ABC with vertices A(3,-6), B(5,-6), and C(7,–9).
Area of a triangle ABC with the given vertices is 3 square units.
Given that, the vertices of a triangle ABC, A(3,-6), B(5,-6), and C(7,–9).
What is the area of triangle formula in coordinate geometry?In Geometry, a triangle is a three-sided polygon that has three edges and three vertices. The area of the triangle is the space covered by the triangle in a two-dimensional plane.
Area of a triangle = [tex]\frac{1}{2} ( |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|)[/tex]
Here, (x1, y1) = A(3,-6), (x2, y2) = B(5,-6), and (x3, y3) = C(7,–9)
Now, the area of a triangle = 1/2 (|3(-6+9)+5(-9+6)+7(-6+6)|)
= 1/2 (|3(3)+5(-3)+7(0)|)
= 1/2 (|(9-15)|)
= 1/2 × 6
= 3 square units
Therefore, area of a triangle ABC with the given vertices is 3 square units.
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If f (x) = x
2 − 2 x , g (x) = x − 2
1) prove that : f(2) = g(2)
2) If g (K) = 7 , find : the value of k
The value of k is 9 for the function g.
To solve this problem we should have a brief concept of algebraic functions.
To solve this problem we have to follow a few steps.
Here f is a function of x and the relation with the function denotes as x²-2x. Also, g is a function of x and the relation with the function denotes as x− 2.
If we put, x = 2 on f(x) = x²-2x. We can write, f(2) = 2²-2.2 = 4 - 4 = 0.
If we put, x = 2 on g (x) = x − 2. We can write, g(2) = x− 2= 2- 2 = 0.
Hence, we can conclude that f(2) = g(2) = 0. ( proved)
Here, g(k) = 7. So, x = k in this relation.
We have to put x= k on g(x) = x− 2 ; now we can write, g (k) = k− 2.
g (k) = k− 2 = 7 as per the question. Therefore k = 7 + 2 = 9
The value of k is 9.
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The correct question is,
If f (x) = x²-2x
g (x) = x − 2
1) prove that : f(2) = g(2)
2) If g (K) = 7, find the value of k
Rectangle ABCD is graphed in the coordinate plane. The following are the vertices of the
rectangle: A(-8,-6), B(-3,-6), C(-3,-4), and D(-8,-4).
Given these coordinates, what is the length of side AB of this rectangle?
The length of side AB of this rectangle is = 5 units
what is rectangle ?
A shape with four straight sides and four angles of 90 degrees ( right angle ) . two of the sides are longer than the other two . a rectangle with four sides of equal length is square .
we could use distance formula to find the length
A (-8,-6) ; B (3 , -6 )
distance =(( X2 - X1 )^2 +( Y2 - Y1 ) ^2 ) ^1/2
AB = (( -3+8)^2+(-6-[-6]^2)^1/2
AB= 5^2
AB = 5 units .
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Evaluate the expression for the given variable.9 - k ÷ 3/4 k=2/3
We are given the following expression:
[tex]9-k\div\frac{3}{4}[/tex]We are also given that k is equal to 2/3. So, we can substitute that into the expression:
[tex]9-\frac{2}{3}\div\frac{3}{4}[/tex]Due to order of operations, we have to do the division first, and then do the subtraction. To do division with fractions, we keep the first fraction the same and take the reciprocal of the second fraction. Then, we can multiply the two fractions. Let's do that:
[tex]9-(\frac{2}{3}\div\frac{3}{4})=9-(\frac{2}{3}*\frac{4}{3})=9-\frac{8}{9}[/tex]Now, we can do the subtraction:
[tex]9-\frac{8}{9}=\frac{81}{9}-\frac{8}{9}=\frac{73}{9}[/tex]Therefore, our answer is 73/9
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Angles are given below.
Define angles.When two straight lines or rays intersect at a single endpoint, an angle is created. The vertex of an angle is the location where two points come together. The Latin word "angulus ," which means "corner," is where the term "angle" originates. When a transversal connects two coplanar lines, alternate interior angles are created. They are located on the transverse sides of the parallel lines, but on the inner side of the parallel lines. At two different locations, the transversal passes through the two lines that are coplanar.
Given,
∠6 and ∠7 = Vertical angles
∠2 and ∠8 = Same side exterior angles
∠1 and ∠5 = Corresponding angles
∠3 and ∠6 = Adjacent angles
∠2 and ∠7 = Alternate exterior angles
∠4 and ∠6 = Same side interior angles
∠1 and ∠2 = Linear pair
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Input x Output y3. -56. -49. -3What is a equation
We are to determine the equation of line by interpreting tabulated results between an independent variable ( x ) and a dependent variable ( y ).
A function is usually expressed as follows:
[tex]y\text{ = f ( x )}[/tex]The above notation gives us the output ( y ) which is a function of input variable ( x ). This means that whatever relationship these two variables have the value of output ( y ) is related to the imput variable ( x ).
We are given a table/list of values of output ( y ) corresponding to each value of input variable ( x ) as follows:
Input ( x ) Output ( y )
3 -5
6 -4
9 -3
There are a series of steps that we must take to arrive at the equation that relates two variables.
Step 1: Determine the type of relationship between two variables by intuition
The first step in the process is the hardest of all. We have to critically analyze each input value ( x ) and its corresponding output value ( y ) with successive pair of values.
There are many types of relationships possible ( polynomial order, exponential, logarithmic, trigonometric, radical, etc .. ).
We can conjure up a way by comparing outputs of successive values to determine the type of relationship possible.
So looking at the first value:
[tex]y\text{ = f ( 3 ) = -5}[/tex]The successive value:
[tex]y\text{ = f ( 6 ) = -4}[/tex]The next successive value:
[tex]y\text{ = f ( 9 ) = -3}[/tex]Here if scrutinize between each successive value of input variable ( x ) we see that there is a "3 unit step-up" in each pair of values i.e ( 3 -> 6 -> 9 ).
Next we compare each output values ( y ) for successive pairs. We see that with every step increase of 3 units in ( x ) value there is an increase of ( 1 ) unit in output value i.e ( -5 -> -4 -> -3 ).
Conclusion: Combing the result of above analysis we see that with each 3 step increase in input value ( x ) there is an increase in output value ( y ) by 1 unit.
This gives us the idea that the two variables are linearly related to one another.
Therefore, the type of relationship is:
[tex]\text{straight line }\text{ }[/tex]Step 2: Recall the equation for the type of relationship between two vairbales x and y
Once we have determined the type of relationship between two variables. We will have to resort to our equation bank and pluck out the corresponding equation that expresses a LINEAR relationship i.e equation of a straight line.
The slope-intercept form of a straight line is:
[tex]y\text{ = m}\cdot x\text{ + c}[/tex]Step 3: Determine the complete equation of function by defining the arbitrary constants.
The above equation is valid for all straight lines that express a linear relationship. However, we seek to find a unique straight line for the given set of points.
Every unique straight line equation would have either of the constants different. The constants defined in a striaght line equation are:
[tex]\begin{gathered} m\colon\text{ The slope( gradient ) of the line} \\ c\colon\text{ The y-intercept} \end{gathered}[/tex]To determine these constants we will use the given pairs of coordinates of input and output variables, x and y respectively.
To determine the slope (m) of an equation:
[tex]m\text{ = }\frac{y_2-y_1}{x_2-x_1}[/tex]The above expression relates the change in output value ( y ) with respect to change in input variable ( x ).
To determine the constant ( m ) we will use the conclusion from Step 1:
"3 step increase in input of ( x ) value there is an increase in output value ( y ) by 1 unit."
Therefore,
[tex]m\text{ = }\frac{+1}{+3}\text{ = }\frac{1}{3}[/tex]To determine the value of y-intercept ( c ). We will plug in the value of ( m ) into the general equation of a straight line written in step 2:
[tex]y\text{ = }\frac{1}{3}x\text{ + c}[/tex]Now, we will use any pair of input and output value.
[tex]x\text{ = 3 , y = -5}[/tex]Substitute the pair of values into the derived equation expressed above and solve for constant ( c ):
[tex]\begin{gathered} -5\text{ = }\frac{1}{3}\cdot(3)\text{ + c} \\ -5\text{ = 1 + c} \\ c\text{ = -6} \end{gathered}[/tex]Note: The above step implies that following equation must satisfy each and every data pair of point given to us ( table ). Or each and every value must lie on the line. For that each value must satisfy the equation of line.
Step 4: Write the complete equation of the relationship
Once we have evaluated the values of equation defining constants ( m and c ). We can simply plug in the values into the general equation relationship ( Linear - slope intercept form ) as follows:
[tex]m\text{ = }\frac{1}{3}\text{ , c = -6}[/tex]Therefore, the equation for the set of values given to us is:
[tex]\begin{gathered} \textcolor{#FF7968}{y}\text{\textcolor{#FF7968}{ = }}\textcolor{#FF7968}{\frac{1}{3}\cdot x}\text{\textcolor{#FF7968}{ - 6}} \\ OR \\ y\text{ = }\frac{x\text{ - 18}}{3} \\ \textcolor{#FF7968}{3y}\text{\textcolor{#FF7968}{ = x - 18}} \end{gathered}[/tex]A quarterback completed 19 out of 30 attempts to pass the football.What was his percent of completion
The percentage of completion is 63.34%.
A grocer wants to mix two kinds of nuts. One kind sells for $2.00 per pound, and the other sells for $2.90 per pound. He wants to mix atotal of 16 pounds and sell it for $2.50 per pound. How many pounds of each kind should he use in the new mix? (Round off the answersto the nearest hundredth.)
Let
x -----> pounds of one kind of nuts ($2.00 per pound)
y ----> pounds of other kind of nuts ($2.90 per pound)
we have that
2x+2.90y=2.50(16) ------> equation 1
x+y=16-----> x=16-y -----> equation 2
solve the system of equations
substitute equation 2 in equation 1
2(16-y)+2.90y=40
solve for y
32-2y+2.90y=40
2.90y-2y=40-32
0.90y=8
y=8.89
Find out the value of x
x=16-8.89
x=7.11
therefore
the answer is
7.11 pounds of one kind of nuts ($2.00 per pound)8.89 pounds of other kind of nuts ($2.90 per pound)A single fair die is tossed. Find the probability of rolling a number greater than 5
Using the probability concept, the odds of rolling a number greater than 5 is 1 :5
What is probability ?
probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur or how likely it is that a proposition is true . the probability of an event is a number between 0 and 1 , where ,roughly speaking ,0 indicates impossibility of the event and 1 indicates certainty.
The odd of a particular experiment is defined thus :
Number of possible outcomes greater than 5 : number of possible below or equal to 5
Sample space = {1, 2, 3, 4, 5, 6}
Outcomes greater than 5 = {6} = 1
Outcomes below or equal to 5 = {1, 2, 3, 4, 5} = 5
The odds equals to 1 : 5
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A laptop computer is purchased for $2100. Each year, its value is 75% of its value the year before. After how many years will the laptop computer be worth$300 or less?
SOLUTION:
After the first year, the price of the laptop computer is;
[tex]P_1=0.75\times2100=1575[/tex]After the second year, the price of the laptop computer is;
[tex]P_2=0.75\times1575=1181.25[/tex]After the third year, the price of the laptop computer is;
[tex]P_3=0.75\times1181.25=885.94[/tex]After the fourth year, the price of the laptop computer is;
[tex]P_4=0.75\times885.94=664.45[/tex]After the fifth year, the price of the laptop computer is;
[tex]P_5=0.75\times664.45=498.34[/tex]After the sixth year, the price of the laptop computer is;
[tex]P_6=0.75\times498.34=373.75[/tex]After the seventh year, the price of the laptop computer is;
[tex]P_7=0.75\times373.75=280.32[/tex]CORRECT ANSWER: 7 years
Give a negation of each inequality.
p < 9
Answer: P can be anything from 8 to below
Example: 8, 7, 6, 5, 4, 3, 2, 1, 0, -1 .....
Solve form. 2m - p = 11f
ANSWER
[tex]m=\frac{11f+p}{2}[/tex]EXPLANATION
We want to solve for m in the equation:
[tex]2m-p=11f[/tex]This means that we want to make m the subject of formula.
That is:
[tex]\begin{gathered} \text{Add p to both sides of the equation:} \\ 2m-p+p=11f+p \\ 2m=11f+p \\ \text{Divide both sides by 2:} \\ \frac{2m}{2}=\frac{11f+p}{2} \\ m=\frac{11f+p}{2} \end{gathered}[/tex]That is the answer.
The graph shows the mass of the bucket containing liquid depends on the volume of liquid in the bucket. Use the graph to find the range of the function.
The graph shows that the range of the function is 0.9 ≤ M ≤ ∞.
Linear FunctionA linear function can be represented by a line. The standard form for this equation is: y=mx+b , for example, y=5x+8.
All functions present their domain and range. The domain of a function is the set of input values for which the function is real and defined. In the other words, when you define the domain, you are indicating for which values x the function is real and defined. While the domain is related to the values of x, the range is related to the possible values of y that the function can have.
From the graph it is possible to see that: the function is a linear function, the values of the coordinate x are represented by the volume (liters) while the values of the coordinate y are represented by the mass (kg).
The question asks for the range of the function. Therefore, you should indicate the possible values of y that the function can have.
For this, you should analyze the axis-y. See that for x=0, the graph shows y =0.9. Therefore, the function starts for values of y >= 0.9 kg. It is possible to verify that when the volume increases, the mass also increases. With this information, you can find that the range is 0.9 ≤ M ≤ ∞.
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Are these lines parallel or not: L1: (1,2), (3,1), and L2: (0,-1), (2,0)
First find the slope of the first line
L1
m = ( y2-y1)/(x2-x1)
m = ( 1-2)/(3-1)
= -1/2
Now find the slope of the second line
L2
m= ( y2-y1)/(x2-x1)
= ( 0 - -1)/(2 -0)
= (0+1)/(2-0)
= 1/2
Parallel lines have the same slope
These lines do not have the same slope, they are different by a negative sign.
These lines are not parallel.
Find in the exact simplified form of an exact expression for the sum of the first n terms of the following series 1+11+111+1111+11111+.... Binary notation is used to represent numbers on a computer. For example, the number 1111 in base two represents 1(2)^3 + 1(2)^2 +1(2)^1+1, or 15 in base ten. (i) Why is the sum above an example of a geometric series? (ii) Which number in base ten is represented by 11 111 111 111 111 111 111 in base two? Explain your reasoning.
Step-by-step explanation:
so, I understand, the given series is written in binary form.
a1 = 1 = 1×2⁰ = 1
a2 = 11 = 1×2¹ + 1× 2⁰ = 3
a3 = 111 = 1×2² + 1×2¹ + 1×2⁰ = 7
a4 = 1111 = 1×2³ + 1×2² + 1×2¹ + 1×2⁰ = 15
a5 = 11111 = 1×2⁴ + 1×2³ + 1×2² + 1×2¹ + 1×2⁰ = 31
...
we see, that
an = 2×(an-1) + 1
a1 = 1
a2 = 2×a1 + 1
a3 = 2×a2 + 1 = 2×(2×a1 + 1) + 1 = 4×a1 + 2 + 1
a4 = 2×a3 + 1 = 2×(2×a2 + 1) = 2×(2×(2×a1 + 1) + 1) + 1 =
= 8×a1 + 2×2 + 2 + 1 = 8×a1 + 7
...
an = (2^(n-1))×a1 + an-1
because
an = 2×(an-1) + 1,
an-1 = (2^(n-1))×a1 - 1
therefore,
an = 2×(2^(n-1))×a1 - 1 = (2^n)×a1 - 1
the sequence of the sums of the first n elements
s1 = a1 = 1
s2 = a1 + a2 = 1 + 3 = 4
s3 = a1 + a2 + a3 = 7 + 3 + 1 = 11
s4 = a1 + a2 + a3 + a4 = 15 + 7 + 3 + 1 = 26
...
(i)
it is NOT a geometric sequence.
for a geometric sequence
an/an-1 = r, and r must be a constant ratio for any n.
but
7/3 = 2.333333...
15/7 = 2.142857143...
these are different, so, the sequence itself is not geometric.
neither is the sequence of the sums of the series. because
11/4 = 2.75
26/11 = 2.363636363...
are different.
1, 2, 4, 8, 16, 32, ... is a geometric sequence (constant r = 2).
but not
1, 3, 7, 15, 31, ...
(ii)
11 111 111 111 111 111 111 in base 2.
the utmost right position is the 2⁰ position. every position further to the left multiples the position value by 2. it is the same process as for numbers in base 10 (just there every position value is multiplied by 10).
we have 6×3 + 2×1 positions = 20 positions.
so, the position values go from 2⁰ to 2¹⁹.
as per the formula for "an" up there, we get
a20 = (2²⁰)×a1 - 1 = 1,048,576 - 1 = 1,048,575