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3 years ago

I need help.............

Mathematics

1 answer:

Grace [21]3 years ago

7 0

$c= b - bx \\bx = b-c\\x= \frac{b-c}{b}$

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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

3 years ago

1 1/3 x 1/14 Help please

nikklg [1K]

Step-by-step explanation:

=11/3×1/14

=3.67×0.07

=0.2568

7 0

2 years ago

Read 2 more answers

which graph represents a line with a slope of -2/3 and a u intercept equal to that of theme u =2/3×-2

Usimov [2.4K]

Answer:

Step-by-step explanation:

6 0

3 years ago

If √(0.09×0.09×x) = 0.09 × 0.09×√z, then the value of x/y is____.

OLEGan [10]

Given that-

√(0.09×0.09×x) = 0.09×0.09×√z

On squaring both sides then

⇛[√(0.09×0.09×x) ]² = (0.09×0.09×√z)²

⇛ 0.09×0.09×x= (0.09)²×(0.09)²×(√z)²

⇛ (0.09)² × x = (0.09)²×(0.09)²×z

⇛ (0.09)² × x/z = (0.09)²×(0.09)²

⇛x/z = (0.09)²×(0.09)²/(0.09)²

⇛x/z = (0.09)²

⇛x/z = 0.09×0.09

⇛x/z = 0.0081

Answer:- Hence the value of x/z will be 0.0081 r respectively.

3 0

2 years ago

Alenkasestr [34]

Answer:

need more characters

8 0

3 years ago