Please help. Answer only if u can show your work or explain

Mathematics

2 answers:

trasher [3.6K]3 years ago

7 0

- 1/2 x + 3 - 4x + 5 + 3/2 x
bring like terms together:-

- 1/2 x - 4x + 3/2 x + 3 + 5 Now you can work out the x's and the numbers
= - 4 1/2 x + 3/2 x + 8
= -4 1/2 x + 1 1/2 x + 8
= -3x + 8

Elenna [48]3 years ago

3 0

Gropu like terms
(-1/2)x+(3/2)x-4x+3+5=
(2/2)x-4x+8=
1x-4x+8=
-3x+8

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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:

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

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

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

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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 $n \geq 0$ ,

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