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3 years ago

Complete the missing number sequence.1/4, 1/2,___, 1, 11/4

Mathematics

1 answer:

mariarad [96]3 years ago

3 0

Answer:If the last number is 1¼ then the answer is ¾

Step-by-step explanation:

From one fraction to the other there is a difference of ¼

¼+¼=½

½+½=¾

¾+¼=1

1+¼=1¼

To identify the difference between the fractions you should get the LCM of the denominator which is 4 then you change all the fractions and make them have the same denominator

½ ,²/4 ,_ ,⁴/4 ,⁵/4

Then work with the numerator

1 , 2, _ ,4 , 5,

The dash is to be filled with 3

So your answer is ¾

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3 years ago

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Find the partial fraction decomposition of the rational expression with prime quadratic factors in the denominator

SpyIntel [72]

$\dfrac{5x^4-7x^3-12x^2+6x+21}{(x-3)(x^2-2)^2}=\dfrac{a_1}{x-3}+\dfrac{a_2x+a_3}{x^2-2}+\dfrac{a_4x+a_5}{(x^2-2)^2}$
$\implies 5x^4-7x^3-12x^2+6x+21=a_1(x^2-2)^2+(a_2x+a_3)(x-3)(x^2-2)+(a_4x+a_5)(x-3)$

When $x=3$ , you're left with

$147=49a_1\implies a_1=\dfrac{147}{49}=3$

When $x=\sqrt2$ or $x=-\sqrt2$ , you're left with

$\begin{cases}17-8\sqrt2=(\sqrt2a_4+a_5)(\sqrt2-3)&\text{for }x=\sqrt2\\17+8\sqrt2=(-\sqrt2a_4+a_5)(-\sqrt2-3)\end{cases}\implies\begin{cases}-5+\sqrt2=\sqrt2a_4+a_5\\-5-\sqrt2=-\sqrt2a_4+a_5\end{cases}$

Adding the two equations together gives $-10=2a_5$ , or $a_5=-5$ . Subtracting them gives $2\sqrt2=2\sqrt2a_4$ , $a_4=1$ .

Now, you have

$5x^4-7x^3-12x^2+6x+21=3(x^2-2)^2+(a_2x+a_3)(x-3)(x^2-2)+(x-5)(x-3)$
$5x^4-7x^3-12x^2+6x+21=3x^4-11x^2-8x+27+(a_2x+a_3)(x-3)(x^2-2)$
$2x^4-7x^3-x^2+14x-6=(a_2x+a_3)(x-3)(x^2-2)$

By just examining the leading and lagging (first and last) terms that would be obtained by expanding the right side, and matching these with the terms on the left side, you would see that $a_2x^4=2x^4$ and $a_3(-3)(-2)=6a_3=-6$ . These alone tell you that you must have $a_2=2$ and $a_3=-1$ .

So the partial fraction decomposition is

$\dfrac3{x-3}+\dfrac{2x-1}{x^2-2}+\dfrac{x-5}{(x^2-2)^2}$

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3 years ago

Simplify: 6y + 5z + 8y – 3z

mezya [45]

Combine like terms. 6y + 8y is 14y. 5z - 3z is 2z. so the answer is 14y + 2z.

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3 years ago

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86,921 in expanded form

Rudik [331]

86921 = 80000 + 6000 + 900 +20 +1

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3 years ago

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Please help me I don’t understand.

attashe74 [19]

Answer:

1 : 64

Step-by-step explanation:

Model:

1/2 in * 1/6 in = 1/12 in^2

Real:

1/3 ft * 1/9 ft = 1/3 * 12 in * 1/9 * 12 in = 16/3 in^2

model : real

1/12 / 16/3 = 1/12 * 3/16 = 3/192 = 1/64

model : real = 1 : 64

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3 years ago

Complete the missing number sequence.1/4, 1/2,___, 1, 11/4​

Complete the missing number sequence.1/4, 1/2,___, 1, 11/4