All categories

4 years ago

If the absolute value of x is greater than 0, then x is greater than 0

Mathematics

1 answer:

lara31 [8.8K]4 years ago

8 0

The absolute value of a number is always positive, so it has to be above zero.

You might be interested in

<img src="https://tex.z-dn.net/?f=%24a%2Ba%20r%2Ba%20r%5E%7B2%7D%2B%5Cldots%20%5Cinfty%3D15%24%24a%5E%7B2%7D%2B%28a%20r%29%5E%7B

riadik2000 [5.3K]

Let

$S_n = \displaystyle \sum_{k=0}^n r^k = 1 + r + r^2 + \cdots + r^n$

where we assume |r| < 1. Multiplying on both sides by r gives

$r S_n = \displaystyle \sum_{k=0}^n r^{k+1} = r + r^2 + r^3 + \cdots + r^{n+1}$

and subtracting this from $S_n$ gives

$(1 - r) S_n = 1 - r^{n+1} \implies S_n = \dfrac{1 - r^{n+1}}{1 - r}$

As n → ∞, the exponential term will converge to 0, and the partial sums $S_n$ will converge to

$\displaystyle \lim_{n\to\infty} S_n = \dfrac1{1-r}$

Now, we're given

$a + ar + ar^2 + \cdots = 15 \implies 1 + r + r^2 + \cdots = \dfrac{15}a$

$a^2 + a^2r^2 + a^2r^4 + \cdots = 150 \implies 1 + r^2 + r^4 + \cdots = \dfrac{150}{a^2}$

We must have |r| < 1 since both sums converge, so

$\dfrac{15}a = \dfrac1{1-r}$

$\dfrac{150}{a^2} = \dfrac1{1-r^2}$

Solving for r by substitution, we have

$\dfrac{15}a = \dfrac1{1-r} \implies a = 15(1-r)$

$\dfrac{150}{225(1-r)^2} = \dfrac1{1-r^2}$

Recalling the difference of squares identity, we have

$\dfrac2{3(1-r)^2} = \dfrac1{(1-r)(1+r)}$

We've already confirmed r ≠ 1, so we can simplify this to

$\dfrac2{3(1-r)} = \dfrac1{1+r} \implies \dfrac{1-r}{1+r} = \dfrac23 \implies r = \dfrac15$

It follows that

$\dfrac a{1-r} = \dfrac a{1-\frac15} = 15 \implies a = 12$

and so the sum we want is

$ar^3 + ar^4 + ar^6 + \cdots = 15 - a - ar - ar^2 = \boxed{\dfrac3{25}}$

which doesn't appear to be either of the given answer choices. Are you sure there isn't a typo somewhere?

7 0

2 years ago

Brainlest!!!

son4ous [18]

Answer:

1st graph: x ≤ -5

2nd graph: x > -5

3rd graph: x ≥ -5

4th graph: x > 5

Step-by-step explanation:

5 0

3 years ago

Read 2 more answers

EMERGENCY! <br><br> How many cubes with side lengths of 1/2 cm does it take to fill the prism?

Sati [7]

Answer:

24 cubes

Step-by-step explanation:

You can figure this a couple of ways.

I usually find it easiest to figure in terms of the number of cubes each dimension represents. The vertical dimension (3/2 cm) is the length of 3 cubes; the front-back dimension (2 cm) is the length of 4 cubes, and the width (1 cm) is the length of 2 cubes.

The total number of cubes required is the product of the dimensions in cube-lengths: 3×4×2 = 24 cubes.

Another way to figure this is to compute the prism volume in the given dimensions (cm³) and the cube volume in the same dimensions, then find the number of cube volumes in the prism volume.

Prism volume = l×w×h = (2 cm)(1 cm)(3/2 cm) = 3 cm³

Cube volume = (1/2 cm)³ = 1/8 cm³

Then the number of cubes that will fit in the prism is ...

(3 cm³)/(1/8 cm³) = 3×8 = 24 . . . . cubes

6 0

4 years ago

Please help me this middle school math i fell asleep help! will give brainliest

castortr0y [4]

Answer:

I think you got it right

Step-by-step explanation:

can you help me Robert Frost's ability to write about the countryside using beautiful imagery was probably influenced by A his life on the farm in the Nem Hampshire countryside. B. his life in the city before he was married C. his keen imagination I need to know this answer

6 0

3 years ago

Find the volume of this right rectangular prism. [Type your answer as a number.]

irakobra [83]

There is no picture

3 0

4 years ago