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3 years ago

Can someone please help? I’m a little stuck here. Factor Differences Of squares.

Mathematics

1 answer:

VikaD [51]3 years ago

5 0

Answer:

29. 9p(2p+1)(2p-1)

31. 4k(2k+5)(2k-5)

Step-by-step explanation:

29. 9p(36p^3/9p-9p/9p)

9p(4p^2-1)

9p((2p)^2-1^2)

9p(2p+1)(2p-1)

31. 4k(16k^3/4k+-100k/4k)

4k(4k^2-25)

4k((2k)^2-5^2)

4k(2k+5)(2k-5)

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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:

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

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

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4 years ago

How do you write 4 1/3 : 7 in simplest form

Tema [17]

4 1/3:7 can be written in many ways
13:21 is the simplest form in whole numbers

Hope this helps! ;)

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3 years ago

One number is 4 more than another, and their sum is 60. What is the smaller number? If x = the larger number and y = the smaller

erik [133]

Answer:

a. $x+y=60$ and $-x+y=4$

Step-by-step explanation:

Let x be the larger number and y be the smaller number.

We have been given that one number is 4 more than another. We can represent this information in an equation as:

$y=x+4...(1)$

We can manipulate this equation as:

$y-x=x-x+4$

$-x+y=4$

We are also told that the sum of both numbers is 60. We can represent this information in an equation as:

$x+y=60...(2)$

Upon looking at our given choices, we can see that option 'a' is the correct choice.

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