Find the area of the parallelogram. (simplify your answer)

<img src="https://tex.z-dn.net/?f=Find%20%7D%20%5Csum%20r%5E%7B2%7D%20%5Ccdot%7B%20%7D%5E%7B20%7D%20C_%7Br%7D%5Cend%7Bequation%7

Serggg [28]

I bet the sum you're referring to is supposed to be

$\displaystyle \sum_{r=0}^{20} r^2 \times {}^{20}C_r$

or equivalently,

$\displaystyle \sum_{r=0}^{20} r^2 \binom{20}r$

where $\binom nk = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.

Recall the binomial series,

$(1+x)^\alpha = \displaystyle \sum_{r=0}^\infty \binom\alpha r x^r$

which is valid for |x| < 1. (Note that if r > α, the binomial coefficient is defined to be zero, so there really are only α many terms when α is a whole number.)

Differentiating both sides with respect to x gives

$\alpha (1+x)^{\alpha-1} = \displaystyle \sum_{r=0}^\infty r \binom\alpha r x^{r-1}$

Multiply both sides by some arbitrary x in |x| < 1 :

$\alpha x (1+x)^{\alpha-1} = \displaystyle \sum_{r=0}^\infty r \binom\alpha r x^r$

Repeat:

$\alpha (1+x)^{\alpha-1} + \alpha(\alpha-1) x(1+x)^{\alpha-2} = \displaystyle \sum_{r=0}^\infty r^2 \binom\alpha r x^{r-1}$

$\alpha x (1+x)^{\alpha-1} + \alpha(\alpha-1) x^2 (1+x)^{\alpha-2} = \displaystyle \sum_{r=0}^\infty r^2 \binom\alpha r x^r$

Let α = 20, and let x approach 1 from below. The right side converges to the sum we want, while the left side converges to

$20 \times 2^{19} + 20\times19\times 2^{18} = (20 + 10\times19)\times2^{19} = \boxed{210\times2^{19}}$

5 0

3 years ago

In 2000, Carl’s Candy Shop sold gummy bears for $0. 38 per pound, and in 2008 they cost $0. 72 per pound. Let y represent the co

tamaranim1 [39]

The linear function that represent the cost per pound of the gummy bears in x years after 2000 is given by:

y(x) = 0.0425x + 0.38

<h3>What is a linear function?</h3>

A linear function is modeled by:

$y = mx + b$

In which:

m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.

b is the y-intercept, which is the value of y when x = 0, which can also be interpreted as the initial value of the function.

In this problem, we have that:

In 2000, Carl’s Candy Shop sold gummy bears for $0. 38 per pound, hence b = 0.38.

In 2008 they cost $0.72 per pound, that is, in 8 years there was an increase of $0.34 per pound, hence the slope is m = 0.34/8 = 0.0425.

Thus, the function is:

y(x) = 0.0425x + 0.38

More can be learned about linear functions at brainly.com/question/24808124

3 0

2 years ago

Please help meeeeee I will mark you!. I need an answer and explanation

kari74 [83]

Answer:

12^2 + x^2 = 13^2

Step-by-step explanation:

The student's error was that he didn't use the Pythagorean Theorem correctly

According to the Pythagorean Theorem, a^2 + b^2 = c^2, where c represents the hypotenuse

However, the student replaced c with x, which isn't the hypotenuse

The hypotenuse is 13, so c should be 13, not x

5 0

3 years ago

:) KE 1/4 - > USING PROPORTIONS TO CHECK... ARE THESE SHAPES SCALE FACTORS? IF YES, WHAT IS THE SCALE FACTOR FROM A TO B? S Р

Viktor [21]

Answer:

What shapes are you talking about?

Step-by-step explanation:

8 0

3 years ago

ILL GIVE YOU BRAINLIEST IF ANSWERED RIGHT!!!! If the quadratic equation has the form y = ax ^2and y = ax ^2+ c, then their axis

Sliva [168]

Answer:

x = 0

Step-by-step explanation:

The axis of the symmetry has the same x coordinate of the vertex

V (x) = -b/2a

in both equations b is equal to 0 and a 0/a number different from 0 gives always 0 as a result

7 0

3 years ago