#1**0 **

To find the sum of f(1) + f(2) + ... + f(40), we can simplify each term and then calculate the sum.

Let's simplify f(n) for any positive integer n:

f(n) = 4n / (sqrt(2n - 1) + sqrt(2n + 1))

To simplify the expression, we can multiply both the numerator and the denominator by the conjugate of the denominator, which is (sqrt(2n - 1) - sqrt(2n + 1)):

f(n) = (4n / (sqrt(2n - 1) + sqrt(2n + 1))) * ((sqrt(2n - 1) - sqrt(2n + 1)) / (sqrt(2n - 1) - sqrt(2n + 1)))

Simplifying further, we get:

f(n) = (4n * (sqrt(2n - 1) - sqrt(2n + 1))) / ((2n - 1) - (2n + 1))

= (4n * (sqrt(2n - 1) - sqrt(2n + 1))) / (-2)

Now we can calculate the sum of f(1) + f(2) + ... + f(40):

sum = f(1) + f(2) + ... + f(40)

= (41(sqrt(21 - 1) - sqrt(21 + 1)))/(-2) + (42(sqrt(22 - 1) - sqrt(22 + 1)))/(-2) + ... + (440(sqrt(240 - 1) - sqrt(240 + 1)))/(-2)

= -2 * (1*(sqrt(21 - 1) - sqrt(21 + 1)) + 2*(sqrt(22 - 1) - sqrt(22 + 1)) + ... + 40*(sqrt(240 - 1) - sqrt(240 + 1)))

Now we can simplify each term in the parentheses and calculate the sum:

sum = -2 * (1*(sqrt(21 - 1) - sqrt(21 + 1)) + 2*(sqrt(22 - 1) - sqrt(22 + 1)) + ... + 40*(sqrt(240 - 1) - sqrt(240 + 1)))

= -2 * (1*(sqrt(1) - sqrt(3)) + 2*(sqrt(3) - sqrt(5)) + ... + 40*(sqrt(79) - sqrt(81)))

We can see that the terms inside the parentheses form a telescoping series, where the square root terms cancel each other out. The only terms that remain are the first and the last: YourTexasBenefits Login

sum = -2 * (sqrt(1) - sqrt(3) + sqrt(3) - sqrt(5) + ... + sqrt(79) - sqrt(81))

= -2 * (sqrt(1) - sqrt(81))

= -2 * (1 - 9)

= -2 * (-8)

= 16

Therefore, the sum of f(1) + f(2) + ... + f(40) is 16.

newsdopJul 13, 2023

#1**+1 **

The equation of the circle can be rewritten in the standard form by completing the square for both the x and y terms:

(x^2 + 4x) + (y^2 - 6y) = -12

To complete the square for the x terms, we need to add (4/2)^2 = 4 to both sides, and for the y terms, we need to add (-6/2)^2 = 9 to both sides:

(x^2 + 4x + 4) + (y^2 - 6y + 9) = -12 + 4 + 9

Simplifying, we have:

(x + 2)^2 + (y - 3)^2 = 1

Comparing this to the standard form of a circle equation, we find that the center of the circle is (-2, 3), and the radius is sqrt(1) = 1.

The distance between the center of the circle (-2, 3) and the point (1, 7) can be found using the distance formula:

distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

= sqrt((1 - (-2))^2 + (7 - 3)^2)

= sqrt(3^2 + 4^2)

= sqrt(9 + 16)

= sqrt(25)

= 5

Therefore, the correct distance between the center of the circle and the point (1, 7) is indeed 5. myccpay

newsdopJun 21, 2023