Answer:
1 time
Step-by-step explanation:
A line is on straight thing that keeps going straight for ever so there for it can only cross the x axis once
(.......) is an example of :?
For this case we have that by definition the distributive property establishes:
[tex](a + b) (c + d) = ac + ad + bc + bd[/tex]
We have the following expression:
[tex](y ^ 2 + y) (x ^ 4 + 3x ^ 3-2x ^ 3) =[/tex]
Applying the distributive property we have:
[tex]y ^ 2 (x ^ 4 + 3x ^ 3-2x ^ 3) + y (x ^ 4 + 3x ^ 3-2x ^ 3)[/tex]
So, the given example is an example of distributive property.
ANswer:
Option B
URGENT NEED HELP WITH A MATH QUESTION
Answer:
The image of point P is (1 , -1)
Step-by-step explanation:
- If point (x , y) rotated about the origin by angle 90° anti-clock wise
∴ Its image is (-y , x)
- If point (x , y) rotated about the origin by angle 90° clock wise
∴ Its image is (y , -x)
- If point (x , y) rotated about the origin by angle 180°
∴ Its image is (-x , -y)
* There is no difference between rotating 180° clockwise or
anti-clockwise around the origin
* Lets solve the problem
∵ P = (-1 , -1)
∵ P is rotated about the origin 90° counterclockwise
- Lets use this rule:
If point (x , y) rotated about the origin by angle 90° counterclockwise
then Its image is (-y , x)
∵ x-coordinate of point P = -1
∴ y-coordinate of the image of point P is -1
∵ y-coordinate of point P is -1
∴ x-coordinate of the image of point P is 1
- Lets write the image of point P
∴ The image of point P is (1 , -1)
What is the following sum 4(5square x^2y)+3(5 square x^2y
Answer:
The answer would be [tex]7(\sqrt[5]{x^{2}y } )[/tex]
Step-by-step explanation: I got it right on Edge 2020
PEOPLE THAT KNOW GEOMETRY HELP A BRO OUT
Answer: Second option.
Step-by-step explanation:
Given the transformation [tex]T:(x,y)[/tex]→[tex](x+3,y+1)[/tex] for the ordered pair (4,3), you can find the preimage point through this procedure:
1) Find the x-coordinate of the preimage point. You know that:
[tex]x+3=4[/tex]
So you must solve for "x":
[tex]x=4-3\\x=1[/tex]
2) Find the y-coordinate of the preimage point. You know that:
[tex]y+1=3[/tex]
So you must solve for "y":
[tex]y=3-1\\y=2[/tex]
Therefore, the preimage point is: (1,2)
If you horizontally stretch the quadratic parent function f(x)=x^2 by a factor of 3. What is the equation
Answer:
G(x)=(1/3x)^2
Step-by-step explanation:
Ap ex
what is the value of negative 1/2 to the fourth power?
A. -16
B. -1/16
C. 1/16
D. 16
Answer: C) 1/16
-1/2 • -1/2 • -1/2 • -1/2 =1/16
[tex]\left(-\dfrac{1}{2}\right)^{-4}=\left(-2\right)^{4}=16[/tex]
NEED HELP WITH A MATH QUESTION
Answer:
[tex]x = 20\°[/tex]
Step-by-step explanation:
By definition we know that the sum of the internal angles of a triangle is always equal to 180 °
in this case we have a right triangle. We know that right triangles have an angle of 90 °.
So the sum of the three sides of this triangle must be equal to 180 °
So
[tex]2x + 10 + 2x + 90 = 180[/tex]
We solve the equation for the variable x.
[tex]4x +100 = 180\\4x = 180-100\\4x = 80[/tex]
[tex]x = 20\°[/tex]
Answer: x=20
Step-by-step explanation:
2x+10+2x+90=180
4x+100=180
4x=80
X=20
The steamboat makes the trips in 5 hours. The yacht makes the boat in 2.5 hours. The yacht is 20 knots faster than the steamboat, the trip time multiplied by the speed equals the distance traveled. You know both boats will travel the same distance. How fast is each boat traveling in knots (nautical miles per hour)?
Answer:
Steam boat = 20 Knots , Yacht = 40 Knots
Step-by-step explanation:
Let the distance travelled by both the boats be x
Case 1: Steamboat
Speed [tex]S_{s}=\frac{x}{5}[/tex]
Case 2 : Yacht
[tex]S_{y}=\frac{x}{2.5}[/tex]
[tex]S_{y}=\frac{2x}{5}[/tex]
Also given that speed of yacht is 20 more than that of steamboat
[tex]S_{y} -S_{s}=20[/tex]
Hence
[tex]\frac{2x}{5} - \frac{x}{5} = 20[/tex]
[tex]\frac{x}{5} = 20[/tex]
[tex]x = 100[/tex]
Hence the distance traveled is 100 naut miles
Hence
Speed of Steam boat = [tex]\frac{100}{5}[/tex] = 20 Knots
Speed of Yacht = [tex]\frac{100}{2.50}[/tex] = 40 Knots
Grading Scale 2 has the following weights- Tests- 60% Quiz- 15% Homework- 10% Final Exam- 15% What do you need to score on the Final Exam to make an 78 in the class if your grades are- Show steps. Test Grades- 85 Quiz- 87 Homework- 68 Final Exam- ??
a. 85(60) + 87(15) + 68(10) + X(15) = 78 X = 49.33
b. 85(6.0) + 87(1.5) + 68(1.0) + X(1.5) = 78 X = 53.62
c. 85(.60) + 87(.15) + 68(.10) + X(.15) = 78 X = 47.67
d. 85(.60) + 87(1.5) + 68(.10) + X(1.5) = 78 X = 35.59
Answer:
c. 85(.60) + 87(.15) + 68(.10) + X(.15) = 78; X = 47.67
Step-by-step explanation:
The weighted average is the sum of products of score and weight. Choice C has the percentage weights properly expressed as decimal values.
___
The other equations could work if the final score of 78 were weighted by the corresponding equivalent of 100%. (X is wrong in the other choices as well.)
__
The attachment shows the "work". Given that choice C is the only correct equation, we presumed that the value of X was correct. The equation solver confirms this. If you want to do it by hand, compute the sum of products, subtract that sum, then divide by 0.15.
X = (78 -70.85)/.15 = 7.15/.15 = 47 2/3
HELP meeeeeeeeeee!!!!!!!!!!
Answer: B
Step-by-step explanation:
Can someone help me do part two please? It’s very important send a picture or something. I don’t even care if you tell me the steps in word form. Please help
1. "Create your own circle on a complex plane."
The equation of a circle in the complex plane can be written a number of ways. For center c (a complex number) and radius r (a positive real number), one formula is ...
|z-c| = r
If we let c = 2+i and r = 5, the equation becomes ...
|z -(2+i)| = 5
For z = x + yi and |z| = √(x² +y²), this equation is equivalent to the Cartesian coordinate equation ...
(x -2)² +(y -1)² = 5²
__
2. "Choose two end points of a diameter to prove the diameter and radius of the circle."
We don't know what "prove the diameter and radius" means. We can show that the chosen end points z₁ and z₂ are 10 units apart, and their midpoint is the center of the circle c.
For the end points of a diameter, we choose ...
z₁ = 5 +5iz₂ = -1 -3iThe distance between these is ...
|z₂ -z₁| = |(-1-5) +(-3-5)i| = |-6 -8i|
= √((-6)² +(-8)²) = √100
|z₂ -z₁| = 10 . . . . . . the diameter of a circle of radius 5
The midpoint of these two point should be the center of the circle.
(z₁ +z₂)/2 = ((5 -1) +(5 -3)i)/2 = (4 +2i)/2 = 2 +i
(z₁ +z₂)/2 = c . . . . . the center of the circle is the midpoint of the diameter
__₁₂₃₄
3. "Show how to determine the center of the circle."
As with any circle, the center is the midpoint of any diameter (demonstrated in question 2). It is also the point of intersection of the perpendicular bisectors of any chords, and it is equidistant from any points on the circle.
Any of these relations can be used to find the circle center, depending on the information you start with.
As an example. we can choose another point we know to be on the circle:
z₄ = 6-2i
Using this point and the z₁ and z₂ above, we can write three equations in the "unknown" circle center (a +bi):
|z₁ - (a+bi)| = r|z₂ - (a+bi)| = r|z₄ - (a+bi)| = rUsing the formula for the square of the magnitude of a complex number, this becomes ...
(5-a)² +(5-b)² = r² = 25 -10a +a² +25 -10b +b²
(-1-a)² +(-3-b)² = r² = 1 +2a +a² +9 +6b +b²
(6-a)² +(-2-b)² = r² = 36 -12a +a² +4 +4b +b²
Subtracting the first two equations from the third gives two linear equations in a and b:
11 -2a -21 +14b = 0
35 -14a -5 -2b = 0
Rearranging these to standard form, we get
a -7b = -5
7a +b = 15
Solving these by your favorite method gives ...
a +bi = 2 +i = c . . . . the center of the circle
__
4. "Choose two points, one on the circle and the other not on the circle. Show, mathematically, how to determine whether or not the point is on the circle."
The points we choose are ...
z₃ = 3 -2iz₄ = 6 -2iWe can show whether or not these are on the circle by seeing if they satisfy the equation of the circle.
|z -c| = 5
For z₃: |(3 -2i) -(2 +i)| = √((3-2)² +(-2-i)²) = √(1+9) = √10 ≠ 5 . . . NOT on circle
For z₄: |(6 -2i) -(2 +i)| = √((6 -2)² +(2 -i)²) = √(16 +9) = √25 = 5 . . . IS on circle
The distance from one corner of a rectangular garden to the other is 13 ft. The length of the garden is 7 ft longer than the width. Write a quadratic equation to find the dimension of the garden. Solve the equation and find the area of the garden in square feet.
Answer:
dimensions: 12 ft by 5 ftarea: 60 ft²Step-by-step explanation:
Let x represent the shorter dimension in feet. Then the longer one is x+7 and the Pythagorean theorem tells us the relation of these to the diagonal is ...
x² + (x+7)² = 13²
2x² +14x + 49 = 169 . . . . eliminate parentheses
x² +7x -60 = 0 . . . . . subtract 169 and divide by 2
(x +12)(x -5) = 0 . . . . factor the equation
x = -12 or +5 . . . . . . . only the positive value of x is useful here.
The short dimension is 5 ft, so the long dimension is 12 ft. The area is their product, 60 ft².
_____
Comment on finding the area
The quadratic equation above can be rearranged and factored as ...
x(x +7) = 60
Since the dimensions of the garden are x and (x+7), this product is the garden's area. This equation tells us the area is 60. We don't actually have to find the dimensions.
In the diagram of triangle LMN, which term describes point P?
Answer:
C. Circumcenter
Step-by-step explanation:
Definitions:
Orthocenter is the point where the three "altitudes" of a triangle meet. An "altitude" is a line that goes through a vertex and is at right angles to the opposite side.
Incenter is the center of an inscribed circle (the circle that fits perfectly inside triangle, just touching all sides). It is where the "angle bisectors" (lines that split each angle in half) meet.
Circumcenter is the center of circumscribed circle. It is where the "perpendicular bisectors" (lines that are at right angles to the midpoint of each side) meet.
Centroid (or center of mass) of a triangle is the point where the three medians of the triangle meet.
As you can see from the definitions, point P is circumcenter, because it is the point of perpendicular bisectors intersection.
Evaluate the step function for the given input values. g(x) = g(2) = g(–2) = g(5) =
Answer:
Step-by-step explanation:
g(2) = g(–2) = g(5) is never true for the step function. One " = " symbol per pair of g values, please.
The step function, which you're calling "g(x)," is 0 from -infinity up to but not including 0. It's 1 from x just greater than 0 through infinity.
Thus:
g(2) = 1 because x is greater than 0.
g(-2) = 0 because x is less than 0.
g(5) = 1 because x is greater than 0.
Answer:
g(2)= 3
g(-2)= -4
g(5)= 5
A surveyor, Toby, measures the distance between two landmarks and the point where he stands. He also measured the angles between the landmarks in degrees.
the triangle has
two sides(65,55)
angles (40,30)
What is the distance, x, between the two landmarks? Round the answer to the nearest tenth.
32.5 m
42.1 m
85.1 m
98.5 m
The Set Up:
x² = (Side1)² + (Side2)² - 2[(Side1)(Side2)]
Solution:
cos(Toby's Angle) • x² = 55² + 65² - 2[(55)(65)] cos(110°)
x² = 3025 + 4225 -7150[cos(110°)]
x² = 7250 - 2445.44x =
√4804.56x = 69.31m
The distance, x, between two landmarks is 69.31m.
Note: The answer choices given are incorrect.
Answer:
98.5 m
Step-by-step explanation:
Refer the attached figure
AB = 55
AD = 65
∠ABC=40°
∠ADC = 30°
We are supposed to find the distance between the two landmarks i.e. BD = BC+CD
In ΔABC
[tex]Cos \theta = \frac{Base}{Hypotenuse}[/tex]
[tex]Cos 40^{\circ} = \frac{BC}{AB}[/tex]
[tex]0.76604444= \frac{BC}{55}[/tex]
[tex]0.76604444 \times 55 =BC[/tex]
[tex]42.132442 =BC[/tex]
In ΔADC
[tex]Cos \theta = \frac{Base}{Hypotenuse}[/tex]
[tex]Cos 30^{\circ} = \frac{CD}{AD}[/tex]
[tex]0.8660254= \frac{CD}{65}[/tex]
[tex]0.8660254 \times 65 =CD[/tex]
[tex]56.291651 =CD[/tex]
So, BD = BC+CD=42.132442+56.291651=98.424≈ 98.5
Hence the distance between the two landmarks is 98.5 m.
How do I find the length of the sides of a right triangle?
Answer:
A^2+B^2=C^2
Step-by-step explanation:
(A)^2+(B)^2+(A)+(B)+2Cos(a)=C^2
Answer:
with a ruler?
Need help with finding the value of x
Answer:
The value of x = 4.1 cm
Step-by-step explanation:
From the figure we can see a circle, with chord 15.6 cm
A right angled triangle with hypotenuse 8.8 cm
To find the value of x
By using Pythagorean theorem,
Base² + Height² = Hypotenuse²
Here Base = x, Height = 15.6/2 = 7.8 cm and Hypotenuse = 8.8 cm
x² + 7.8² = 8.8²
x² = 8.8² - 7.8²
= 77.4 - 60.84
= 16.56
x = √16.56 = 4.07 ≈ 4.1 cm
Therefore the value of x = 4.1 cm
A line is drawn through (–7, 11) and (8, –9). The equation y – 11=-4/3 (x + 7) is written to represent the line. Which equations also represent the line? Check all that apply.
Answer:
[tex]y=-\frac{4}{3}x+\frac{5}{3}[/tex]
[tex]4x+3y=5[/tex]
Step-by-step explanation:
we have
[tex]y-11=-\frac{4}{3}(x+7)[/tex] -----> equation of the line into point slope form
step 1
Find the equation of the line into slope intercept form
[tex]y=mx+b[/tex]
where
m is the slope
b is the y-intercept (value of y when the value of x is equal to zero)
For x=0
[tex]y-11=-\frac{4}{3}(0+7)[/tex]
[tex]y=-\frac{28}{3}+11[/tex]
[tex]y=\frac{5}{3}[/tex]
therefore
[tex]b=\frac{5}{3}[/tex]
substitute
[tex]y=-\frac{4}{3}x+\frac{5}{3}[/tex] ----> equation of the line into slope intercept form
step 2
Find the equation of the line in standard form
[tex]Ax+By=C[/tex]
we have
[tex]y=-\frac{4}{3}x+\frac{5}{3}[/tex]
Multiply by 3 both sides
[tex]3y=-4x+5[/tex]
[tex]4x+3y=5[/tex] ---> equation of the line in standard form
what is the value of c such that x^2-20x+c is a perfect-square trinomial?
Answer:
c=100
Step-by-step explanation:
To complete the square, it would really help if the coefficient of x^2 is 1 which is so formula for c in this case is just (b/2)^2
So (-20/2)^2
simplifying gives
(-10)^2
100
c=100
Answer:
100
Step-by-step explanation:
The question is on making the equation a perfect square
Given ;
[tex]x^2 - 20x + c[/tex]
To get c;
[tex]c=(\frac{b}{2} )^2[/tex]
where ;
[tex]b= -20[/tex]
[tex]c= (\frac{-20}{2} )^2 = 10^2 = 100[/tex]
A number is called a "decreasing number" if each digit in the number is less than the digit to its left. For example, 87420 is a decreasing number. How many five-digit decreasing numbers are there?
Answer:
252
Step-by-step explanation:
The question is fully equivalent to asking how many subsets of length 5 there are of 10 objects (digits 0–9). That number is 10C5, where nCk is the number of ways to choose k objects from a list of 10. The value of that is ...
nCk = n!/(k!(n-k)!)
There are 252 ways to choose 5 numbers from the digits 0-9:
10C5 = 10!/(5!(10-5)!) = 10·9·8·7·6/(5·4·3·2·1) = 9·4·7 = 252
_____
The order of the selection doesn't matter, because the selected digits are always arranged in decreasing order to form a decreasing number.
Answer:
Could you please post a video or give more simple explanations to someone who does not have prior knowledge of such problems?
Step-by-step explanation:
Graph the following piecewise function.
Shown below
Step-by-step explanation:A piecewise function is a function defined by two or more equations. In this problem, we have a function defined by two equations.
First, a quadratic equation:
[tex]x^2+2[/tex]
This is a parabola that opens upward and starts at the point [tex](-5,27)[/tex] and whose vertex is the point [tex](0,2)[/tex]. Keep in mind that [tex]x=-5[/tex] is included in the domain of the function, and we know this by the symbol ≤ that includes the equality.
Second, a linear equation:
[tex]x-4[/tex]
The graph of this is a linear function with slope [tex]m=1[/tex] and starts at the point [tex](3,-1)[/tex]. Keep in mind that [tex]x=3[/tex] is included in the domain of the function by the same symbol ≤
Moreover, at [tex]x=3[/tex] there is a jump discontinuity.
Which of the following constants can be added to x 2 - x to form a perfect square trinomial?
A) 1/4
B) 1/2
C) 1
What is the circumference of circle P?
Express your answer in terms of .
AB = 14 in.
Answer:
[tex]C=14\pi\ in[/tex]
Step-by-step explanation:
we know that the circumference is equal to
[tex]C=\pi D[/tex]
we have
[tex]D=14\ in[/tex]
substitute
[tex]C=\pi (14)[/tex]
[tex]C=14\pi\ in[/tex]
Answer:
Circumference of given circle = 14π in
Step-by-step explanation:
Points to remember
Circumference of circle = 2πr
Where 'r' is the radius of circle
To find the circumference of circle
Here diameter AB = 14 in
radius, r = 14/2 = 7 in
Circumference = 2πr
= 2 * * 7
= 14π in
Therefore the correct answer is,
Circumference = 14π in
For the following sequence determine the common difference (if it is an arithmetic sequence) or the common ratio (if it is a geometric sequence).
Answer:
the common ratio is 1/√7
Step-by-step explanation:
The differences are not constant, but the ratio is:
7/(7√7) = 1/√7
√7/7 = 1/√7
The common ratio is 1/√7.
The given sequence, 7√7, 7, √7, ..., is a geometric sequence with a common ratio of √7.
A geometric sequence is a sequence of numbers in which each term is equal to the previous term times a constant value, called the common ratio. In the given sequence, we can see that each term is equal to the previous term times √7. For example, the second term, 7, is equal to the first term, 7√7, times √7. The third term, √7, is equal to the second term, 7, times √7. And so on.
To find the common ratio of a geometric sequence, we can divide any term of the sequence by the previous term. For example, we can divide the second term, 7, by the first term, 7√7, to get:
7 / 7√7 = √7
We can also divide the third term, √7, by the second term, 7, to get:
√7 / 7 = √7
In both cases, we get the same answer, √7. This tells us that the common ratio of the geometric sequence is √7.
Therefore, the answer is common ratio = √7.
To learn more about geometric sequence here:
https://brainly.com/question/33243139
#SPJ3
The graphs of f(x)=2x+2 and g(x)=2(2)^x are shown.
What are the solutions to the equation 2x+2=2(2) ^2 ?
Select each correct answer.
0
1
2
4
Answer:
{0, 1}
Step-by-step explanation:
The x-coordinates of the points of intersection are 0 and 1. These are the solutions to f(x)=g(x).
The solution to the given system of equations is [0, 1].
What is the solution of two equations?
A solution of a system (two equations) in two variables is an ordered pair that makes both the equations true, that solution corresponds to the intersection point of the two equations.
According to the given question.
We have a graph for the two equations [tex]2x+2[/tex] and [tex](2)2^{x}[/tex].
From the graph we can se that both the equations or a system of equations are coinciding from 2 to 4. And for 2 to 4 the x-coordinate is 0 to 1.
Therefore, the solution to the given system of equations is [0, 1].
Find out more information about solution to the system of equations:
https://brainly.com/question/9351049
#SPJ2
Find the tenth term in the following geometric sequence. 8, 4, 2, 1, . . .
a) 13
b) 0.0078
c) 0.0156
d) 12.5
Answer:
c) 0.0156
Step-by-step explanation:
The general term a[n] of a geometric sequence is given in terms of the first term a[1] and common ratio r as ...
a[n] = a[1]r^(n-1)
The given sequence has an initial term of a[1]=8 and a common ratio of 4/8=1/2. Then the general term is ...
a[n] = 8(1/2)^(n-1)
The 10th term is then ...
a[10] = 8(1/2)^(10-1) = 8(1/2)^9 = 8/512
a[10] = 0.015625 ≈ 0.0156
triangle CRV has side lengths that measure 10 centimeters, 12 centimeters, and 15 centimeters. Which of the following best describes this type of triangle?
A) scalene triangle
b) equilateral triangle
c) isosceles triangle
d) obtuse triangle
Answer:
A. Scalene, which just means three unequal sides.
Step-by-step explanation:
An equilateral triangle has three equal sides and an isosceles triangle two, so it's not those.
Obtuse means there's at least one obtuse angle, always opposite the longest side. Here we have 10^2+12^2=244 which is greater than 15^2=225, indicating the angle opposite 15 is acute, so not this one either.
Answer:
A) Scalene triangle.
Step-by-step explanation:
We have been given that triangle CRV has side lengths that measure 10 centimeters, 12 centimeters, and 15 centimeters. We are asked to determine the type of triangle CRV.
We can see that the given measure of all sides are different. We know that a triangle is known a scalene triangle, when all its sides has different measures.
Since all the sides of triangle CRV has different measure, therefore, triangle CRV is a scalene triangle and option A is the correct choice.
Using the given points, determine Δy.
(-3, -5) and (0, 10)
A. Δy = 3
B. Δy = 5
C. Δy = 13
D. Δy = 15
[tex]\bf (\stackrel{x_1}{-3}~,~\stackrel{y_1}{-5})\qquad (\stackrel{x_2}{0}~,~\stackrel{y_2}{10}) \\\\\\ slope = m\implies \cfrac{\stackrel{\Delta y}{ y_2- y_1}}{\stackrel{\Delta x}{ x_2- x_1}}\implies \cfrac{10-(-5)}{0-(-3)}\implies \cfrac{10+5}{0+3}\implies \cfrac{\stackrel{\stackrel{\Delta y}{\downarrow }}{\boxed{15}}}{3}[/tex]
BRAINLIEST
write the algebraic expression
"the sum of 22 and four times a number"
The answer would be:
Let n be "a number"
22 + 4n
Hope this helped!
~Just a girl in love with Shawn Mendes
Answer:
22+4x
Step-by-step explanation:
The number should be x as a letter symbol.
The sum should be add sign.
22+4x is the correct answer.
I hope this helps you, and have a wonderful day!
1. If the domain of a function f(x) is the set {10, 20, 30}. What does the information tell you about f-1(x)?
2. If the graph of a function f(x) includes the point (3, 0), what point must the graph of f-1(x) include? Explain.
3.The first term in an arithmetic sequence is 2. The twelfth term is 211. Find the value of n so that an = 135.
PART A (1 in diagram)
Each mapping diagram represents a function because none of the elements in the inputs maps on to two different elements in the outputs.
1. The domain of the given function f(x) is the set {10, 20, 30}.
All we can say about
[tex] {f}^{ - 1} (x)[/tex]
is that, it has a range of {10, 20, 30}.
This is because the domain of a function becomes the range of its inverse function.
2. If the graph of a function f(x) includes the point (3, 0), then the graph of the inverse function ,
[tex] {f}^{ - 1} (x) [/tex]
will contain the point (0,3).
Reason:The point on the graph of the inverse function is obtained by reflecting the corresponding point on the graph of the function in the line y=x. We just have to swap the coordinates.
In simple terms the domain of the function becomes the range of the inverse function and vice versa.
3. If the first term in an arithmetic sequence is 2.
Then we have a=2.
If the twelfth term is 211, then
[tex]a + 11d = 211...(1)[/tex]
Put a=2 into this equation.
[tex]2 + 11d = 211[/tex]
Solve for d.
[tex]11d = 211 - 2[/tex]
[tex]11d = 209[/tex]
[tex]d = \frac{209}{11} = 19[/tex]
If
[tex]a_n = 135[/tex]
then
[tex]a + (n - 1)d = 135[/tex]
Substitute a=2 and d=19
[tex]2 + (n - 1)19 = 135[/tex]
[tex]19(n - 1) = 133[/tex]
[tex](n - 1) = \frac{133}{19} [/tex]
[tex]n - 1 = 7[/tex]
[tex]n = 7 + 1 = 8[/tex]
Therefore n=8