MATHEMATICS HIGH SCHOOL

Evaluate\rm (3^2+1)/(3^2-1)+(5^2+1)/(5^2-1)+(7^2+1)/(7^2-1)+\ldots+(101^2+1)/(101^2-1) =
With step by step explanation !​

Answers

Answer 1
Answer:

It's easier to deal with the symbolic sum (in sigma notation),

\displaystyle\sum_(k=1)^(50)((2k+1)^2+1)/((2k+1)^2-1)

Expanding the terms in the fraction, computing the quotient, and decomposing into partial fractions gives

((2k+1)^2+1)/((2k+1)^2-1) = (4k^2 + 4k + 2)/(4k^2 + 4k)

=\frac12*(2k^2 + 2k + 1)/(k^2 + k)

=\frac12\left(2+\frac1{k(k+1)}\right)

=\frac12\left(2 + \frac1k - \frac1{k+1}\right)

and it's the latter two terms that reveal a telescoping pattern.

In case you need more details about the partial fraction decomposition, we are looking for coefficients a and b such that

\frac1{k(k+1)}=\frac ak+\frac b{k+1}

or

1 = a(k+1) +bk =(a+b)k+a

which gives a = 1, and a + b = 0 so that b = -1.

Our sum has been rearranged as

\displaystyle\frac12\sum_(k=1)^(50)\left(2+\frac1k-\frac1{k+1}\right)=\sum_(k=1)^(50)1+\frac12\sum_(k=1)^(50)\left(\frac1k-\frac1{k+1}\right)=50+\frac12\sum_(k=1)^(50)\left(\frac1k-\frac1{k+1}\right)

The remaining telescoping sum is

1/2 [(1/1 - 1/2) + (1/2- 1/3) + (1/3- 1/4) + … + (1/48- 1/49) + (1/49- 1/50) + (1/50 - 1/51)]

and you can see how there are pairs of numbers that cancel, so that the sum reduces to

1/2 [1/1 - 1/51] = 1/2 [1 - 1/51] = 1/2 × 50/51 = 25/51

So, our original sum ends up being

\displaystyle\sum_(k=1)^(50)((2k+1)^2+1)/((2k+1)^2-1) = 50 + (25)/(51) = \boxed{(2575)/(51)}

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Find the area of the green sector if m∠ A = π 6 and the circle has a radius of 10 inches.

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The area of the sector, when the measure of the central angle is in radians, can be solved using this equation:
A = (n/2) r²

From the given values,
A = (π/6/2) (10²)
A = 25π/6 = 26.18 in²

Therefore, the area of the sector with the central angle measuring π/6 with the radius of the circle 10 inches is 26.18 in²

The set of integers between -4 and 2?

Answers

-3,-2,-1,0,1 are all between -4 and 2 excluding those numbers.

A candy store keeps track of its sales during the month of October. Below is the number of sales for the first, second, third and fourth week of the month. Week 1 ; 2 ; 3 ; 4
Candy Sales 97 ; 168 ; 259 ; 375

Which graph could represent the data shown below?

Answers

The graph of the given values is required.

Option B is correct.

The points are

(1,97)[tex]</p><p>[tex](2,168)

(3,259)

(4,375)

Let us find if the points are in the form of a line by finding the slope.

(168-97)/(2-1)=71

(259-168)/(3-2)=91

71\neq 91

So, the function is not linear.

Therefore, A and C are incorrect.

Also the function is not decreasing so option D is also eliminated.

Hence, option B is correct.

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From the attached diagram you can see that regression that best fits given data is quadratic (not linear).

  • option 1 is false, because it gives linear regression;
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  • options 3 and 4 are false, because graphs are decreasing.

Answer: correct choice is B.

which of the following group of numbers are all prime numbers a.2,5,15,19 b.7,17,29,49 c.3,11,23,31 d.2,3,5,9,

Answers

Prime numbers are those that cannot be divided by another number (other than themselves and 1). In other words, prime numbers have factors of themselves and 1.

For example, 3 is a prime number because its only factors are 3 and 1.

6 is not a prime number because it can be factored to 3*2.

Therefore, the correct answer is c. 3,11,23,31
Prime numbers can only be divided by 1 and itself.

First group:- It contains 15, and 15 can be divided by 3 and 5 to get a whole number.

Second group:- 49 can be divided by 7 to receive 7. So this is wrong.

Third group:-  This group can be considered all prime numbers, so this is correct.

Fourth group:- Incorrect. Because 9/3 = 3

Answer:-   c. 3,11,23,31 are all prime numbers. 

What temperature in Fahrenheit is 50 degree celsius? (Enter numeric value only. If rounding is necessary, round to the nearest whole number.)

Answers

Step #1: Subtract 32. Step #2: Mutliply by 5. Step #3: Divide by 9... 50-32 = 18; 18 * 5 = 90; 90 / 9 = 10 degrees celsius.

The Length of a standard jewel case is 7cm more than its width. The area of the rectangular top of the case is 408cm. Find the length and width of the jewel Case

Answers

Answer:

The length of the case is 24 cm and its width is 17cm.

Step-by-step explanation:

The Length of a standard jewel case is 7cm more than its width.

Let the length be represented by L and the width be represented by W, this means that:

L = 7 + W

The area of the rectangular top of the case is 408cm². The area od a rectangle is given as:

A = L * W

Since L = 7 + W:

A = (7 + W) * W = 7W + W²

The area is 408 cm², hence:

408 = 7W + W²

Solving this as a quadratic equation:

=> W² + 7W - 408 = 0

W² + 24W - 17W - 408 = 0

W(W + 24) - 17(W + 24) = 0

(W - 17) (W + 24) = 0

=> W = 17cm or -24 cm

Since width cannot be negative, the width of the case is 17 cm.

Hence, the length, L, is:

L = 7 + 17 = 24cm.

The length of the case is 24 cm and its width is 17cm.
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