Solve for x (-4x + 4).-3= 24

Prove or disprove (from i=0 to n) sum([2i]^4) <= (4n)^4. If true use induction, else give the smallest value of n that it doe

ddd [48]

Answer:

The statement is true for every n between 0 and 77 and it is false for $n\geq 78$

Step-by-step explanation:

First, observe that, for n=0 and n=1 the statement is true:

For n=0: $\sum^{n}_{i=0} (2i)^4=0 \leq 0=(4n)^4$

For n=1: $\sum^{n}_{i=0} (2i)^4=16 \leq 256=(4n)^4$

From this point we will assume that $n\geq 2$

As we can see, $\sum^{n}_{i=0} (2i)^4=\sum^{n}_{i=0} 16i^4=16\sum^{n}_{i=0} i^4$ and $(4n)^4=256n^4$ . Then,

$\sum^{n}_{i=0} (2i)^4 \leq(4n)^4 \iff \sum^{n}_{i=0} i^4 \leq 16n^4$

Now, we will use the formula for the sum of the first 4th powers:

$\sum^{n}_{i=0} i^4=\frac{n^5}{5} +\frac{n^4}{2} +\frac{n^3}{3}-\frac{n}{30}=\frac{6n^5+15n^4+10n^3-n}{30}$

Therefore:

$\sum^{n}_{i=0} i^4 \leq 16n^4 \iff \frac{6n^5+15n^4+10n^3-n}{30} \leq 16n^4 \\\\ \iff 6n^5+10n^3-n \leq 465n^4 \iff 465n^4-6n^5-10n^3+n\geq 0$

and, because $n \geq 0$ ,

$465n^4-6n^5-10n^3+n\geq 0 \iff n(465n^3-6n^4-10n^2+1)\geq 0 \\\iff 465n^3-6n^4-10n^2+1\geq 0 \iff 465n^3-6n^4-10n^2\geq -1\\\iff n^2(465n-6n^2-10)\geq -1$

Observe that, because $n \geq 2$ and is an integer,

$n^2(465n-6n^2-10)\geq -1 \iff 465n-6n^2-10 \geq 0 \iff n(465-6n) \geq 10\\\iff 465-6n \geq 0 \iff n \leq \frac{465}{6}=\frac{155}{2}=77.5$

In concusion, the statement is true if and only if n is a non negative integer such that $n\leq 77$

So, 78 is the smallest value of n that does not satisfy the inequality.

Note: If you compute $(4n)^4- \sum^{n}_{i=0} (2i)^4$ for 77 and 78 you will obtain:

$(4n)^4- \sum^{n}_{i=0} (2i)^4=53810064$

$(4n)^4- \sum^{n}_{i=0} (2i)^4=-61754992$

7 0

3 years ago

<img src="https://tex.z-dn.net/?f=%5Csqrt%7B%5C%5C%5Cfrac%7B9%2B%5Csqrt%7B17%7D%20%7D%7B2%7D" id="TexFormula1" title="\sqrt{\\\f

Alborosie

$\huge\underline\pink{✏ANSWER}$

<h2>Hello and sorry but it's a Gatheted Question Can you fix it Can you? </h2>

Sorry for answering a nonsense

#BrainliestBunch

4 0

2 years ago

Which number line represtents the solution to |x+4|=2?

vlada-n [284]

Answer:

Points -2 and -6 on the number line are the two solutions.

Step-by-step explanation:

Use the definition of absolute value as a starting point

$|x|=x\,\,\mbox{for}\,\,x\geq 0\\|x|=-x\,\,\mbox{for}\,\,x$

To solve the equation, you need to treat the two cases as above:

$|x+4|=x+4=2\,\,\,\mbox{for}\,\,x+4\geq 0\implies x\geq -4\\x+4=2\implies x=-2$

The solution x=-2 is consistent with the condition x>=-4, so it is the first and valid solution. Now the second case of the absolute value:

$|x+4|=-(x+4)=2\,\,\,\mbox{for}\,\,x+4< 0\implies x$

Again, the second solution -6 complies with the requirement that x<-4, so it is valid.

7 0

2 years ago

Let K and T be the current ages of two siblings, Katie and Thomas. Katie is currently twice the age of Thomas. In 6 years, Katie

Pavlova-9 [17]

Answer:

Katie is 6 years old and Thomas is 3 years old

Step-by-step explanation:

Given that we should let K and T be the current ages of two siblings, Katie and Thomas.

If Katie is currently twice the age of Thomas then,

K = 2T

and in 6 years, Katie will be 4 times Thomas's current age then

K + 6 = 4T

Solving both equations simultaneously by substituting the value of K given in the first equation into the second

2T + 6 = 4T

Collect like terms

6 = 4T - 2T

6 = 2T

Divide both sides by 2

T = 3

Recall that K = 2T

K = 2 * 3

= 6

Hence Katie is 6 years old while Thomas is 3 years old

6 0

2 years ago

What is the percent decline, to the nearest whole number, of rusty patched bumble bees from 20 years ago to today?

Andrej [43]

Sasuke is pretty cool

6 0

3 years ago

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