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2 years ago

PROBLEM 6

Mathematics

1 answer:

Inessa [10]2 years ago

8 0

Answer:

See below.

Step-by-step explanation:

The nth triangular number is Tn = n(n + 1)/2, the nth square number Qn = n^2 and the nth pentagonal number Pn = (3n^2 - n) / 2.

(a. Q6 + P5 = 6^2 + (3(5)^2 - 5)/2

= 36 + 35 = 71.

3T5 + Q5 + 1 = 3* 5(5 + 1) / 2 + 5^2 + 1

= 90 / 2 + 26

= 71.

So Q6 + P5 = 3T5 +Q5 +1.

(b) Qn+1 + Pn = (n + 1)^2 + (3n^2 - n) / 2

= n^2 + 2n + 1 + (3n^2 - n) / 2

= n^2 + 2n + 1 + 1.5n^2 - 0.5n

= 2.5n^2 + 1.5n + 1.

3Tn +Qn + 1 = 3n(n + 1) / 2 + n^2 + 1

= 3n^2 + 3n / 2 + n^2 + 1

= 1.5n^2 + 1.5n + n^2 + 1

= 2.5n^2 + 1.5n + 1.

Therefore they are equal for all positive n.

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Find the area of triangle DEF with vertices D(-1,2), E(3,-2) and F(-2,-2).*

Veronika [31]

Answer:

<em>The area of the triangle DEF is 10.</em>

Step-by-step explanation:

<u>Area of a Triangle:</u>

A triangle of base B and height H (both must be perpendicular) has an area of:

$\displaystyle A=\frac{BH}{2}$

The image below shows the triangle formed by the points DEF. It's important to notice the points E and F lie on the same horizontal line because they both have the same y-coordinate.

This fact simplifies the calculations since we can easily compute the length the of base as the difference of their x-coordinates:

B=3-(-2)=3+2=5

Being the base of a horizontal line, the height of the triangle can be calculated as the difference of y-coordinates of D and the height of that line.

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With these two values, we calculate the area:

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3 years ago

Explain the importance of factoring.

ivolga24 [154]

Answer:

Factoring is a useful skill in real life. Common applications include: dividing something into equal pieces, exchanging money, comparing prices, understanding time, and making calculations during travel.

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3 years ago

Use the regression calculator to compare the teams’ number of runs with their number of wins. a 2-column table with 9 rows. colu

-Dominant- [34]

The equation of the trend line is ŷ = 0.15x - 23.21

<h3>How to determine the average rate of change of the trend line?</h3>

The points are given as:

hr =>808, 768, 655, 684, 637, 619, 613, 609, 563

w => 93, 94, 66, 81, 86, 75, 61, 69, 55.

Using a graphing calculator, we have the following calculation summary:

Sum of X = 5956
Sum of Y = 680
Mean X = 661.7778
Mean Y = 75.5556
Sum of squares (SSX) = 50569.5556
Sum of products (SP) = 7547.1111

The regression equation is represented as:

ŷ = bX + a

Where:

b = SP/SSX

b = 7547.11/50569.56

b = 0.14924

a = MY - bMX

a = 75.56 - (0.15*661.78)

a = -23.20961

This gives

ŷ = 0.14924X - 23.20961

Approximate

ŷ = 0.15x - 23.21

Hence, the equation of the trend line is ŷ = 0.15x - 23.21