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3 years ago

I'm not good at math I how wild I work it out

Mathematics

1 answer:

a_sh-v [17]3 years ago

5 0

The probability of landing on red the first time is 1/2, because there are two colors on the wheel. The probability of landing on blue the second time in that individual round is 1/2. The probability of landing on red the third time in that individual round is 1/2.

Multiply 1/2 by 1/2 by 1/2

Answer: The probability is 1/8

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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

Paul wants to know when gas would have cost about $1.50 per gallon. What step should he take?

EastWind [94]

he should know how much gallons he need to fill his car up
multiply 1.50 by the amount of gallons he need so he should know how much money to spend on that many gallons
hoped i helped and brainlist plz

3 0

3 years ago

HELP FAST 100 POINTS 100 POINTS What is the value of the expression when a = 5, b = 4 , and c = 6?⁢ 4a2b−c 2 5 9 10

ohaa [14]

For this case we must find the value of the following expression:

$4a ^ 2b-c$