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It has been shown that [tex]3P_{n} =T_{3n-1}[/tex], [tex]3P_{n} =T_{3n-1}[/tex] and hence [tex]3P_{n} =T_{3n-1}[/tex].

Given: [tex]T_{n}[/tex] is the nth triangular number. [tex]Q_{n}[/tex] is the nth square number. [tex]P_{n}[/tex] is the nth pentagonal number.

To show:

Triangular numbers are numbers formed by addition of consecutive natural numbers starting from 1. Square numbers are numbers formed by squaring natural numbers. Pentagonal numbers are numbers formed by distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex.

The formula for each of them is given by:

[tex]T_{n}=\frac{n(n+1)}{2}[/tex]

[tex]Q_{n}=n^{2}[/tex]

[tex]P_{n}=\frac{3n^{2}-n}{2}[/tex]

(a).

Now, put [tex]n=3n-1[/tex] in [tex]T_{n}=\frac{n(n+1)}{2}[/tex] to get,

[tex]T_{3n-1}=\frac{(3n-1)(3n)}{2}[/tex]

[tex]T_{3n-1}=3\frac{n(3n-1)}{2}[/tex]

[tex]T_{3n-1}=3\frac{3n^{2}-n}{2}[/tex]

Put [tex]P_{n}=\frac{3n^{2}-n}{2}[/tex] in [tex]T_{3n-1}=3\frac{3n^{2}-n}{2}[/tex] to get,

[tex]T_{3n-1}=3P_{n}[/tex]

So, [tex]3P_{n}=T_{3n-1}[/tex]

(b).

Now,

[tex]P_{n}-Q_{n}=\frac{3n^{2}-n}{2}-n^{2}[/tex]

[tex]P_{n}-Q_{n}=\frac{3n^{2}-n-2n^{2}}{2}[/tex]

[tex]P_{n}-Q_{n}=\frac{n^{2}-n}{2}[/tex]

Put [tex]n=n-1[/tex] in [tex]T_{n}=\frac{n(n+1)}{2}[/tex] to get,

[tex]T_{n-1}=\frac{(n-1)(n-1+1)}{2}[/tex]

[tex]T_{n-1}=\frac{(n-1)n}{2}[/tex]

[tex]T_{n-1}=\frac{n^{2}-n}{2}[/tex]

Comparing [tex]P_{n}-Q_{n}=\frac{n^{2}-n}{2}[/tex] and [tex]T_{n-1}=\frac{n^{2}-n}{2}[/tex], we get,

[tex]P_{n}-Q_{n}=T_{n-1}[/tex]

Now, put [tex]n=3n[/tex] in [tex]P_{n}-Q_{n}=T_{n-1}[/tex] to get,

[tex]P_{3n}-Q_{3n}=T_{3n-1}[/tex]

As shown in part (a), [tex]T_{3n-1}=3P_{n}[/tex]

Put [tex]T_{3n-1}=3P_{n}[/tex] in [tex]P_{3n}-Q_{3n}=T_{3n-1}[/tex] to get,

[tex]P_{3n}-Q_{3n}=3P_{n}[/tex]

[tex]P_{3n}-3P_{n}=Q_{3n}[/tex]

So, [tex]P_{n}-Q_{n}=T_{n-1}[/tex] and hence [tex]P_{3n}-3P_{n}=Q_{3n}[/tex]

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Answer:

20×10=200mm

70:200

70/10=7

200/10=20

7:20 is your answer in it's simplest form

Answer:

5 dollors i just tookthe test 4

Step-by-step explanation:

Arianna bought 0 bags of oranges. It's possible that she may have bought other items along with the oranges or there might be a pricing discrepancy.

To find out how many oranges Arianna bought, we can use the given price of a bag of 4 oranges and the total amount she spent on oranges.

Let's set up a proportion:

Price of a bag of 4 oranges = $2.84

Amount Arianna spent on oranges = $3.55

Let "x" be the number of bags of oranges Arianna bought.

Since each bag contains 4 oranges, the total number of oranges she bought is 4 times the number of bags (4x).

Now, we can set up the proportion:

(Price of a bag of 4 oranges) / (Amount spent on oranges) = (Number of bags bought) / 4x

$2.84 / $3.55 = 1 / 4x

To solve for "x," cross-multiply:

$2.84 * 4x = $3.55 * 1

11.36x = 3.55

Now, solve for "x":

x = 3.55 / 11.36

x ≈ 0.3125

Since we can't buy a fraction of a bag, we round the value of "x" to the nearest whole number:

x ≈ 0 (rounded to the nearest whole number)

So, Arianna bought 0 bags of oranges. It's possible that she may have bought other items along with the oranges or there might be a pricing discrepancy.

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Answer:i honestly dont know just trying to get points

Step-by-step explanation:

Answer:

your will becanswer $95.00

Answer:

105

Step-by-step explanation:

instalment time 10 months

so from 1050÷10

105 ans

Answer:

90=30+(x+10). remove blanket

90=30+x+10. correct like terms

90=30+10+x

90=40+x. 40 will cross the = and change from +ve to -ve sign

90-40=x.

therefore

x=40

Answer:

x = 50

Step-by-step explanation:

The 2 angles sum to 90° , that is

x + 10 + 30 = 90

x + 40 = 90 ( subtract 40 from both sides )

x = 50

Answer:

70

Step-by-step explanation:

she burns 250 calories per hour .to burn 3500 calories and shed 1 pound she has to work 14 hours and to shed 5 pounds she has to work 14×5=70 hours

Answer:

[tex]6 \sqrt{5} [/tex]

Step-by-step explanation:

Take out Square root 5 as common and then add numbers

[tex]4 \sqrt{5} + 2 \sqrt{5} \\ 6 \sqrt{5} [/tex]

Answer:

We reach at 8.

Step-by-step explanation:

The starting point is 2 .

Jump at the six places, the final point reaches at 8, so the point is 8.

Answer: Answer C , Only student 2, because dilation preserve angle measures and side lengths are proportional

The true statement is only student 2, because dilations preserve angle measures and side lengths are proportional.

Transformation of geometrical figures or points is the manipulation of a given figure to some other way.

Different types of transformations are Rotation, Reflection, Glide reflection, Translation and Dilation.

Dilation is a type of transformation where the figure is enlarged or made smaller such that it preserves the shape but not size.

In Reflection the figure is flipped. In other words, a figure when undergoes reflection becomes it's mirror image.

Dilation preserves the angle measures, so the triangles will be similar.

Reflection, on the other hand, will result in congruent triangles.

Student 1 said that  Reflect △ABC across the y-axis and then dilate it by a scale factor of 1 with the center of dilation at the origin.

If the scale factor is 1, then the dilated image will be the same as the original image.

So the resulting triangle will be congruent to the original triangle.

In the case of what student 2 said, scale factor is 2. Triangle is enlarged.

So the resulting image will be similar to the original one.

Hence student 2 is correct.

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Answer:

40

Step-by-step explanation:

The first step is to calculate the base area of the prism

= 15×30

= 450

The volume is then calculated as follows

= 450×h

= 450h

The capacity is 18 liters

= 18×1000

= 18,000

Therefore the height of the rectangular prism can be calculated as follows

450h= 18,000

h= 18,000/450

= 40

Hence the height is 40 cm

Hope this may help you ☺

Answer:

29%

Step-by-step explanation:

From the two way table given : the relative frequency of boys among those who cannot bike to school ;

Here, we are only concerned with thilose who cannot Bikento school and not all of the data :

The relative frequency of boys among those who cannot bike to school is :

Number of boys who can't bike to school / total number of people who can't bike to school

Number of boys who can't bike = 4

Total who cannot bike = 14

Hence, 4 /14 = 0.2857142

0.2857142 * 100% = 28.57% = 29% (nearest percent)

Hope this helps!! :)))))))

Answer:

$27.00

Step-by-step explanation:

We first have to start at the base value of $45.00, as without that, we have nothing to go off of. We can work right to left because we start at the rightmost point of the problem.

Therefore, we start with finding 9/10 of 45.00 . This is as simple as multiplying 9/10 with 45.00, resulting in 40.5

To figure out a percentage relative to real numbers, we first have to turn that percentage into a fraction or decimal. To turn it into a decimal, we simply divide by 100, and 66 2/3 divided by 100 is roughly 0.6666 , or 2/3 . Multiplying our result of 40.5 by this, we get $27.00 as our answer.

Answer:

3.78 × [tex]10^{2}[/tex] = 378

Step-by-step explanation:

A survey is conducted in order to collect a piece of specific information. The number of people that can be considered as a sample for the given population is 320.

A survey is conducted in order to collect a piece of specific information from a group of people chosen from the entire population who denotes the entire population for those questions.

Given the population consist of 3200 people, therefore, the number of people that can be considered as a sample for the given population is 320.

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Answer:

ygftyhhhhgffyujhhgyjjjhhguj

Answer:

D

Step-by-step explanation:

Answer: one solution

this is all i know

Answer:

D

Step-by-step explanation:

Given

1 - [tex]\frac{1}{2}[/tex] (6x - 4) = 3(1 - x) ← distribute parenthesis on both sides

1 - 3x + 2 = 3 - 3x

3 - 3x = 3 - 3x

Since both sides of the equation are the same , then

This indicates the equation has infinitely many solutions.

Answer:

Y is equal to 2

Step-by-step explanation:

X equals -8

So,

4(-8) + 9y = -14

-32 + 9y = -14

-32 + 32 + 9y = -14 + 32

9y = 18

1/9 × 9y = 18 × 1/9

y = 2

Answer:

use the pythagorean theorem

Step-by-step explanation:

a²+b²=c²

Answer:

i)

Odd numbers:

P(odd and odd) = 3/6*3/6 = 1/4

ii)

Total outcomes:

Sum of 8:

Sum of 12:

P(8 or 12) = 5/36 + 1/36 = 6/36 = 1/6

iii)

Prime numbers:

Odd numbers:

P(prime or odd) = 3/6*3/6 + 3/6*3/6 = 1/4 + 1/4 = 1/2

Answer:

D   (0, 3)

Step-by-step explanation:

To go from B (8, 3) to C (3, 8)

subtract 5 from x, add 5 to y

A (5, -2)

subtract 5 from x,   x = 0

add 5 to y,   y = 3

D (0, 3)

Answer:

Step-by-step explanation:

The diagonals of a rhombus divides it into four congruent right triangles.  Each triangle is a mirror image of the adjacent.

Therefore angle 2 is the alternate interior angle of 57°, and angle 3 is the mirror image, so also 57°.

Angles 1 is the complement of angle 3, and angle 4 is equal to the complement of angle 2 , so both equal 90-57 = 33°.

To summarize,

angle 1 = 33°

angle 2 = 57°

angle 3 = 57°

angle 4 = 33°

Answer:

5/7 or 0.71 is the answer

Answer:

[tex]a_{n}[/tex] = 2n + 5

Step-by-step explanation:

There is a common difference between consecutive terms, that is

9 - 7 = 11 - 9 = 13 - 11 = 15 - 13 = 2

This indicates the sequence is arithmetic with nth term

[tex]a_{n}[/tex] = a₁ + (n - 1)d

where a₁ is the first term and d the common difference

Here a₁ = 7 and d = 2 , then

[tex]a_{n}[/tex] = 7 + 2(n - 1) = 7 + 2n - 2 = 2n + 5

Answer:

295/72

Step-by-step explanation:

Answer:

[tex] { ({x}^{2}. {y}^{6} )}^{ \frac{1}{8} } \\ {x}^{2 \times \frac{1}{8} } . {y}^{6 \times \frac{1}{8} } \\ = {x}^{ \frac{2}{8} } . {y}^{ \frac{6}{8} } \\ = {x}^{ \frac{1}{4} } . {y}^{ \frac{3}{4} } [/tex]