What is 3.1. Divided by 4

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

4 years ago

A student earned a grade of 80% on a math test that had 20 problems. How many problems on this test did the student answer corre

boyakko [2]

Answer:

16 problems.

Step-by-step explanation:

100% divided by 20 questions.

100/20 is equal to 5, therefore each question is worth 5 points.

Grade of 80% divided by points per question.

80/5 is 16 so the student answered 16 questions correctly.

6 0

3 years ago

Read 2 more answers

What is greater than 52 I need to fill in the blank with 2 more numbers

Alona [7]

Answer:

Numbers greater than 52 include but are not limited to 53, 53, 55, up to numbers like 123,786, 638,908, and 100,000,000

8 0

3 years ago

Given an exponential function for compounding interest, A(x) = P(.96)x, what is the rate of change?

uranmaximum [27]

Answer:

The rate of change is positive 96%

Step-by-step explanation:

.96 = 96%

4 0

3 years ago

Complete the table by supplying the measure of inscribed angle and its intercepted arc

Nadusha1986 [10]

Answer/Step-by-step explanation:

Based on the inscribed angle theorem, any inscribed angle = ½ of its intercepted arc

Thus:

✔️Inscribed angle = 57°

Intercepted arc would be: 2*57 = 114°

✔️Intercepted arc would be: 123°

Inscribed angle = ½(123) = 61.5°

✔️Intercepted arc would be: 46°

Inscribed angle = ½(46) = 23°

✔️Inscribed angle = 69°

Intercepted arc would be: 2*69 = 138°

✔️Intercepted arc would be: 155°

Inscribed angle = ½(155) = 77.5°

3 0

3 years ago