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liberstina [14]
3 years ago
13

Identify solutions to each inequality- pls someone help me

Mathematics
1 answer:
pantera1 [17]3 years ago
6 0

Answer:

a and b

Step-by-step explanation:

since the inequality says that all numbers less than 11 are the answer a and b are the choices that are below 11

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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
P+q=1 and pq=-20. What are the values of p and q?​
andrey2020 [161]

Answer:

Step-by-step explanation:

5, -4

5-4=1

5(-4)=-20

6 0
2 years ago
Your family goes to a Southern-style restaurant for dinner. There are 7 people in your family. Some order the chicken dinner for
butalik [34]

Answer:

a) 4 people ordered streak dinner

b) 3 people ordered chicken dinner

Step-by-step explanation:

Let the number of people that ordered steak dinner be x

and the number of people that ordered chicken dinner be y

From the question,

There are 7 people in your family

Hence, we can write that

x + y = 7 ...... (1)

Also from the question

Some order the chicken dinner for $13 and some order the steak dinner for $18 and the total bill was $111.

We can then write that

13y + 18x = 111 ...... (2)

From equation (1)

x + y = 7

Then,

x = 7 - y ...... (3)

Put this into equation (2)

13y + 18x = 111

13y + 18(7 - y) = 111

13y + 126 - 18y = 111

-5y = -15

5y = 15

y = 15 / 5

y = 3

Put the value of y into equation (3) to get x

x = 7 - y

Then

x = 7 - 3

x = 4

Hence, 4 people ordered streak dinner

and 3 people ordered chicken dinner

a) 4 people ordered streak dinner

b) 3 people ordered chicken dinner

3 0
2 years ago
How can the answer me 9x9x8x7 if you are not supposed to repeat digits?
user100 [1]

Answer: 4,536

Step-by-step explanation: Multiply your two 9's to get 81. Multiply your 8 and 7 to get 56. Multiply 81 and 56 together to get your answer of 4,536.

9 x 9 = 81

8 x 7 = 56

81 x 56 = 4,536

Use long multiplication to evaluate this equation.

I hope this helps you out! ♥

4 0
3 years ago
Solve for c in a+b+c=180
AleksAgata [21]

Answer:

c = -a - b + 180

Step-by-step explanation:

Isolate the c. Note the equal sign, what you do to one side, you do to the  other. Subtract a and b from both sides

a (-a) + b (-b) + c = 180 (-a) (-b)

c = 180 - a - b

c = -a - b + 180 is your answer

~

4 0
3 years ago
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