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2 years ago

I need help finding x please?

Mathematics

2 answers:

Art [367]2 years ago

5 0

If we create a circle around the line it will equal 360. Half of the circle must equal 180. For the straight line we have 125 and 30. To get 180 we need a certain number. 180-155=25. The number that we are searching for is 25.

enot [183]2 years ago

3 0

$\large\mathfrak{{\pmb{\underline{\blue{To\:find}}{\blue{:}}}}}$

The value of $x$ .

$\large\mathfrak{{\pmb{\underline{\orange{Solution}}{\orange{:}}}}}$

$\longrightarrow{\green{x\:=\: 25° }}$

$\large\mathfrak{{\pmb{\underline{\red{Step-by-step\:explanation}}{\red{:}}}}}$

We know that,

$\sf\purple{Sum\:of\:angles\:on\:a\:straight\:line\:=\:180°}$

➪ 125° + $x$ + 30° = 180°

➪ $x$ + 155° = 180°

➪ $x$ = 180° - 155°

➪ $x$ = 25°

Therefore, the value of $x$ is 25°.

Now, the three angles of the triangle are 125°, 25° and 30°.

$\large\mathfrak{{\pmb{\underline{\pink{To\:verify}}{\pink{:}}}}}$

✒ 125° + 25° + 30° = 180°

✒ 180° = 180°

✒ L. H. S. = R. H. S.

$\boxed{Hence\:verified.}$

$\huge{\textbf{\textsf{{\orange{My}}{\blue{st}}{\pink{iq}}{\purple{ue}}{\red{35}}{\green{ヅ}}}}}$

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2 years ago

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 $n \geq 0$ ,

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