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3 years ago

Need help as soon as possible

Mathematics

1 answer:

levacccp [35]3 years ago

8 0

2nd answer
(The inverse of the original conditional statement)

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<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

A box has dimensions of 14 inches long, 1.5 feet wide, and 7 inches high. What is the volume of the box? The formula for the vol

goblinko [34]

convert the feet to inches

1 foot = 12 inches

so 1.5 ft =1.5*12 = 18 inches

now multiply all 3

14*18*7 = 1764 cubic inches

8 0

3 years ago

The diagram shows a regular polygon<br> What is the value of x?<br> How do I solve this

I am Lyosha [343]

Answer: x=120°

Step-by-step explanation:

What you are looking for is the value of the exterior angle of the regular polygon.

For a n-sided regular polygon, the external angle is given by the formula 360°/n.

Because the shape in this example is a regular triangle, the value of n is 3.

360°/n

=360°/3

=120°

Therefore, x=120°

3 0

3 years ago

Adison earned $25 mowing her neighbors lawn. Then she loaned her friend $18 and got $50 from her grandmother. She now has $86. H

Marat540 [252]

29 dollars because 25-18+50=57 and 86-57=29

4 0

3 years ago

PIPEILY<br> A. 4(3x - 5)<br> 12x - 24

zloy xaker [14]

Answer:

No solution

Step-by-step explanation:

The question is 4(3x-5)=12x-24

so we do the step:12x-20=12x-24

No solution.

3 0

3 years ago

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