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3 years ago

Help me asap I don’t know what to do

Mathematics

1 answer:

sattari [20]3 years ago

3 0

Answer:

J 8.5 >=2.5p

Step-by-step explanation:

To do this plug in 3.4 for each version of p. The only one that would be true would be J.

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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

4 years ago

How do you convert fractions to a specific base with exponents? Is there a specific way or equation for this? For example, i nee

Lilit [14]

Answer:

$\frac{1}{2} = {2}^{ - 1}$

Step-by-step explanation:

remember the property

${a}^{ - n} = \frac{1}{ {a}^{n} }$

then u get equation with same base

8 0

3 years ago

Which statements describes this trapozoid?

Anna11 [10]

A trapezoid/trapezium is a quadrilateral (4 sided shape) with exactly one pair of parallel sides.

It is a quadrilateral (a closed plane shape with four linear sides) that has at least one pair of parallel lines for sides

4 0

3 years ago

Find an ordered pair (x,y) that is solution to the equation x-3y=9

Rashid [163]

(0,-3) , (9,0) , (12,1)

7 0

3 years ago

Please help i don't know what I'm doing!!! :( multiply and simplify cot x( sin x - sec x)

fgiga [73]

Answer:

cos(x) -csc(x)

Step-by-step explanation:

It is helpful to know the relations between the trig functions:

$\cot{x}=\dfrac{\cos{x}}{\sin{x}}\\\\\sec{x}=\dfrac{1}{\cos{x}}\\\\\csc{x}=\dfrac{1}{\sin{x}}$

Then the given expression can be simplified as follows:

$\cot{x}(\sin{x}-\sec{x})=\dfrac{\cos{x}}{\sin{x}}\left(\sin{x}-\dfrac{1}{\cos{x}}\right)=\cos{x}-\dfrac{1}{\sin{x}}\\\\=\boxed{\cos{x}-\csc{x}}$

6 0

3 years ago