240 = 2 × 120
120 = 2 × 60
60 = 2 × 30
30 = 2 × 15
15 = 3 × 5
The 2, 2, 2, 2, 3 and 5 are all the prime factors of 240. So, 2^4 × 3 × 5

1500 = 3 × 500
500 = 5 × 100
100 = 5 × 20
20= 5 × 4
4 = 2 × 2
The 3, 5, 5, 5, 2 and 2 are all the prime factors of 1500. So, 2² + 3 + 5³

4 0

3 years ago

Read 2 more answers

<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

How much coffee will each person get if 3 people share 1/2 a pound of coffee?

Usimov [2.4K]

Answer:6 pounds you need to divide 3 by .5 and you will get 5

Step-by-step explanation:

5 0

3 years ago

A manufacturer of lawnmowers wanted to know the most important product attributes to consumers. The company's research team deve

stepladder [879]

Answer: Ordinal measurement scale

Step-by-step explanation:

In ordinal measurement scale the order of values or variables are very important. Just as in the case above, the rating from 1-3 is according to the order of it's importance.

Thanks

7 0

3 years ago