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Leno4ka [110]
3 years ago
10

The sum of the first n positive odd numbers is square.

Mathematics
1 answer:
ololo11 [35]3 years ago
3 0

Answer:

See explanation below

Step-by-step explanation:

All the odd numbers can be written as 2n-1 where n is an integer.

The sum (S) of the first n positive odd numbers can be written as:

S=1+3+5+.........+(2n-5)+(2n-3)+(2n-1)

If we change the order to a decreasing order we would write it this way:

S=(2n-1)+(2n-3)+(2n-5)+.........+5+3+1

Let's sum up both equations, we are going to pair the first number of our first equation with the first number of our second equation, we can see that each pair sums up 2n ( let's see this: (2n-1 + 1 =2n, 2n-3 + 3 = 2n, 2n-5 + 5 = 2n...)) therefore we would get n pairs that sum up 2n.

2S=n(2n)\\2S=2n^{2} \\S=n^{2}

Therefore, the sum of the first n positive odd numbers is square.

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2×-20/3 =2x help me please ​
Llana [10]

Answer:

x = - 5

Step-by-step explanation:

Given

\frac{2x-20}{3} = 2x ( multiply both sides by 3 to clear the fraction )

2x - 20 = 6x ( subtract 2x from both sides )

- 20 = 4x ( divide both sides by 4 )

- 5 = x , that is x = - 5

4 0
3 years ago
Aaron bought a set of plastic ice cubes with a mixture of water and air inside.
LekaFEV [45]

Answer:

Volume of water and air is 3,000 mm³

Step-by-step explanation:

Given:

Length of base = 15 mm

Width of base = 20 mm

Height of figure = 30 mm

Find:

Volume of water and air is inside each of the plastic ice cubes.

Computation:

Height of each rectangular pyramid = Height of figure / 2

Height of each rectangular pyramid = 30 / 2

Height of each rectangular pyramid = 15 mm

Volume of each rectangular pyramid = lbh / 3

Volume of each rectangular pyramid = [15 × 20 × 15] / 3

Volume of each rectangular pyramid = [4,500] / 3

Volume of each rectangular pyramid = 1,500 mm³

Volume of water and air is inside each of the plastic ice cubes = 2 × Volume of each rectangular pyramid

Volume of water and air is inside each of the plastic ice cubes = 2 × 1,500

Volume of water and air is inside each of the plastic ice cubes = 3,000 mm³

6 0
3 years ago
Find the missing part.
alexgriva [62]

X^2= AB^2 + BC^2

X^2= 8^2 + 10^2

X^2= 64+100

X^2= 164

X= 12.80

5 0
3 years ago
A line that includes the point (0, 9) has a slope of 9. What is its equation in slope-intercept
Rufina [12.5K]
Slope intercept: y = mx + b
Y = 9x + b, plug in point
9 = 9(0) + b, b = 9
Solution: y = 9x + 9
7 0
3 years ago
Set up the characteristic equation, which is given by r2 − 5r − 6 = 0. The roots of this equation are given by r1 = 6 and r2 = −
s344n2d4d5 [400]

Answer:

Step-by-step explanation:

From your characteristic equation, your recursive equation is

f(n+2) = 5 f(n=1) +6f(n)

the general solution:

f(n) = A6^n + B(-1)^n

The initial conditions are

f(0) =1 ~~~and~~~ f(1) = 3

For f(0) = 1, that is

f(0) = A6^0 + B(-1)^0 = A+ B=1 (*)

For f(1) = 3, that is

f(1) = A6^1 + B(-1)^1 = 3

6A- B = 3 (**)

From (*) and (**) you solve for A and B

you have A = 4/7 and B= 3/7

Replace A, B into the general one, you have the particular solution for the given condition

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4 0
3 years ago
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