Solve the quadratic equation by factoring:

(b) If a series follows geometric progression, show that the ratio of the sum of term to the sum from (n+1) term to (2n) term is

Nadusha1986 [10]

Proof with explanation:

We know that the sum of first 'n' terms of a Geometric progression is given by

$S_{n}=\frac{a(1-r^{n})}{1-r}$

where

a = first term of G.P

r is the common ratio

'n' is the number of terms

Thus the sum of 'n' terms is

$S_{n}=\frac{a(1-r^{n})}{1-r}$

Now the sum of first '2n' terms is

$S_{2n}=\frac{a(1-r^{2n})}{1-r}$

Now the sum of terms from $(n+1)^{th}$ to $(2n)^{th}$ term is $S_{2n}-S_{n}$

Thus the ratio becomes

$\frac{S_{n}}{S_{2n}-S_{n}}\\\\=\frac{\frac{a(1-r^{n})}{1-r}}{\frac{a(1-r^{2n})}{1-r}-\frac{a(1-r^{n})}{1-r}}\\\\=\frac{1-r^{n}}{r^{n}-r^{2n}}\\\\=\frac{1-r^{n}}{r^{n}(1-r^{n})}\\\\=\frac{1}{r^{n}}$

5 0

3 years ago

Julie bought three circular rugs to put in her bedroom. Each rug has a 4 ft radius. What is the area of her floor that will be c

V125BC [204]

Answer is D.) 48π ft²

Given:
3 circular rugs.
radius: 4 ft

Area of a circle = π r²
A = 3.14 * (4ft)²
A = 3.14 * 16ft²
A = 50.24 ft²

Total area of 3 rugs

50.24 ft² * 3 rugs = 150.72 ft²

Equation leaving π.

150.72ft² ÷ 3.14 = 48ft²

D.) 48π ft²

4 0

3 years ago

Read 2 more answers

How do you expressed the difference in the measures of center as a multiple of the measure of variation in a box plot when the I

pishuonlain [190]

You just use a calculator

3 0

2 years ago

What is the first step to solving the division problem below?

Otrada [13]

Answer/Step-by-step explanation:

Work your way left to right solving each one going by the rules of PEMDAS

Have a nice day and do a good job, best of luck :)

6 0

3 years ago

Read 2 more answers

Would someone please help me solve these questions?~

olasank [31]

Question # 17

Solution:

Graphing the vertices of a triangle and transforming by first translating

(x, y) → (x -3, y - 3) and then reflecting in the line y = x.

As A(-1, 1), B(0, 3) and C(3, 4)

After Translation: (x, y) → (x - 3, y - 3)

A(-1, 1) → A'(-4, -2)

B(0, 3) → B'(-3, 0)

C(3, 4) → C'(0, 1)

After Reflection: in the line y = x

The reflection of the point (x, y) across the

line y = x is the point (y, x).

A'(-4, -2) → A''(-2, -4)

B'(-3, 0) → B''(0, -3)

C'(0, 1) → C''(1, 0)

Pleas check the top right side of the attached figure a.

Is the image same if the order of the transformation is switched ? (Reflection, then glide). Changing the order of transformation.

A(-1, 1), B(0, 3) and C(3, 4) 

 After Reflection: in the line y = x

The reflection of the point (x, y) across the line y = x is the point (y, x).

A(-1, 1) → A'(1, -1)

B(0, 3) → B'(3, 0)

C(3, 4) → C'(4, 3)

After Translation: (x, y) → (x - 3, y - 3)

A'(1, -1) → A''(-2, -4)

B'(3, 0) → B''(0, -3)

C'(4, 3) → C''(1, 0)

No, the image is no longer the same if order of the transformation is switched as shown at the right bottom of figure a.

Answering the YES and NO Questions:

<h3>Is the composition a glide reflection?</h3>

No, the composition is not a glide reflection as glide reflection is commutative. If we reverse the direction of the composition, the outcome will be affected.

Check the figure at top right side of figure a, and compare it with the figure at the bottom right side of figure a. They are not the same.

<h3></h3><h3>Is the image same if the order of the transformation is switched ? (Reflection, then glide)</h3>

No, the image is no longer the same if order of the transformation is switched as shown at the right bottom figure of figure a.

The top right side of figure a was transformed after first translation of (x, y) → (x - 3, y - 3) and then doing reflection across the line y = x. While the bottom right side of figure a was made by first reflection across the line y= x and then translation of (x, y) → (x - 3, y - 3. So, after changing the order both figures do not remain the same.

Question # 18

Solution:

Graphing the vertices of a triangle and transforming by first 90 rotating clockwise and then reflection in the y axis.

A(-1, 1), B(0, 3) and C(3, 4)

After transforming by first 90 rotating clockwise about the origin. The new position of point (x, y) will be (y, -x)

A(-1, 1) → A'(1, 1)

B(0, 3) → B'(3, 0)

C(3, 4) → C'(4, -3)

After Reflection: in the line y axis

The reflection of the point (x, y) in the line line y-axis is the point (-x, y).

A'(1, 1) → A''(-1, 1)

B'(3, 0) → B''(-3, 0)

C'(4, -3) → C''(-4, -3)

Please check the top right side of figure b.

If the order of the transformation were performed in reverse order ? (Reflection, then rotation), would the final image by the same?

A(-1, 1), B(0, 3) and C(3, 4)

After Reflection: in the line y-axis

The reflection of the point (x, y) in the line y-axis is the point (-x, y).

A(-1, 1) → A'(1, 1)

B(0, 3) → B'(0, 3)

C(3, 4) → C'(-3, 4)

After transforming by first 90 rotating clockwise about the origin. The new position of point (x, y) will be (y, -x)

A'(1, 1) → A''(1, -1)

B'(0, 3) → B''(3, 0)

C'(-3, 4) → C''(4, 3)

No, the image is no longer the same if order of the transformation is switched as shown at the right bottom of figure b.

Answering the YES and NO Questions:

If the transformation were performed in reverse order (reflection. then rotation), would the final image be the same?

No, if we reverse the direction of the composition, the final image would not be same.

Check the figure at top right side of figure b, and compare it with the at the bottom right side of figure b. They are not the same.

Would the reflection of triangle ABC in the line y = x have produced the same final image?

Yes, the reflection of triangle ABC in the line y = x would have produced the same final image. Compare the figure of the bottom left with the bottom right of figure b.

As A(-1, 1), B(0, 3) and C(3, 4)

After Reflection: in the line y = x

The reflection of the point (x, y) across the

line y = x is the point (y, x).

A(-1, 1) → A''(1, -1)

B(0, 3) → B''(3, 0)

C(3, 4) → C''(4, 3)

Hence, the reflection of triangle ABC in the line y = x would have produced the same final image. Compare the figure of the bottom left with the bottom right of figure b.

Keywords: transformation, reflection, rotation, translation, triangle

Learn more about different transformations from brainly.com/question/11914738

#learnwithBrainly

3 0

3 years ago