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3 years ago

7

A painter used 20 drops of red paint to make a certain shade of pink.How many drops of white paint should he add to 12 drops of

red paint to make the same shade?

1 answer:

BartSMP [9]3 years ago

5 0

He have to use 27 drops of red <span>paint to make it
</span>

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Determine whether the sequences converge.

Alik [6]

$a_n=\sqrt{\dfrac{(2n-1)!}{(2n+1)!}}$

Notice that

$\dfrac{(2n-1)!}{(2n+1)!}=\dfrac{(2n-1)!}{(2n+1)(2n)(2n-1)!}=\dfrac1{2n(2n+1)}$

So as $n\to\infty$ you have $a_n\to0$ . Clearly $a_n$ must converge.

The second sequence requires a bit more work.

$\begin{cases}a_1=\sqrt2\\a_n=\sqrt{2a_{n-1}}&\text{for }n\ge2\end{cases}$

The monotone convergence theorem will help here; if we can show that the sequence is monotonic and bounded, then $a_n$ will converge.

Monotonicity is often easier to establish IMO. You can do so by induction. When $n=2$ , you have

$a_2=\sqrt{2a_1}=\sqrt{2\sqrt2}=2^{3/4}>2^{1/2}=a_1$

Assume $a_k\ge a_{k-1}$ , i.e. that $a_k=\sqrt{2a_{k-1}}\ge a_{k-1}$ . Then for $n=k+1$ , you have

$a_{k+1}=\sqrt{2a_k}=\sqrt{2\sqrt{2a_{k-1}}\ge\sqrt{2a_{k-1}}=a_k$

which suggests that for all $n$ , you have $a_n\ge a_{n-1}$ , so the sequence is increasing monotonically.

Next, based on the fact that both $a_1=\sqrt2=2^{1/2}$ and $a_2=2^{3/4}$ , a reasonable guess for an upper bound may be 2. Let's convince ourselves that this is the case first by example, then by proof.

We have

$a_3=\sqrt{2\times2^{3/4}}=\sqrt{2^{7/4}}=2^{7/8}$
$a_4=\sqrt{2\times2^{7/8}}=\sqrt{2^{15/8}}=2^{15/16}$

and so on. We're getting an inkling that the explicit closed form for the sequence may be $a_n=2^{(2^n-1)/2^n}$ , but that's not what's asked for here. At any rate, it appears reasonable that the exponent will steadily approach 1. Let's prove this.

Clearly, $a_1=2^{1/2}$ . Let's assume this is the case for $n=k$ , i.e. that $a_k$ . Now for $n=k+1$ , we have

$a_{k+1}=\sqrt{2a_k}$

and so by induction, it follows that $a_n$ for all $n\ge1$ .

Therefore the second sequence must also converge (to 2).

4 0

3 years ago

Please help me with this i need help she is gonna call my mom again!!!

telo118 [61]

Answer:

1. 15

2. -23

3. -16

Step-by-step explanation:

1. Find h(-8)

h(t) = $\frac{5}{4}(4 - t)$

h(-8) = $\frac{5}{4}(4 -(-8))$

h(-8) = $\frac{5}{4}(4+8) =\frac{5}{4}(12)=5(3)=15$

2. What is the value of g(-3)

g(x) = 4(x - 5) + $x^{2}$

g(-3) = 4(-3 - 5) $-3^{2}$

g(-3) = 4(-8) + 9

g(-3) = -32 + 9

g(-3) = -23

3. Find f(10)

f(x) = 14 - 3x

f(10) = 14 - 3(10)

f(10) = 14 - 30

f(10) = -16

7 0

3 years ago

Read 2 more answers

Please answer correctly !!!! Will mark brainliest !!!!!!!!!!!!

vfiekz [6]

Answer:

I believe that your answer is a

Step-by-step explanation:

I remember that the exponent would be multiplied by both sides.

6 0

3 years ago

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My life is helpless.... could you please help? i cant even understand this I'm only 10.

Nadya [2.5K]

I think the equation is y= -1/3x+1. The slope is going down, so it’s negative. And the y intercept is 1. And how I got a fraction is by using rise/run.

5 0

2 years ago

There were 15 apples in a bowl. Dan used some apples to make a pie. Now there are 7 apples in the bowl. How many apples did Dan

Leni [432]

15 - 7 = 8 So Dan used 8 apples so there is 7 apples left

5 0

3 years ago

Read 2 more answers