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SSSSS [86.1K]
3 years ago
8

I've solved most! Just need help with 9, 10, 14.

Mathematics
1 answer:
Ugo [173]3 years ago
7 0
7. Correct

- - -

9. By the ratio test, the series will converge if

\displaystyle\lim_{n\to\infty}\left|\frac{\frac{b^{n+1}(x-a)^{n+1}}{\ln(n+1)}}{\frac{b^n(x-a)^n}{\ln n}}\right|

The limit reduces to

\displaystyle|b(x-a)|\lim_{n\to\infty}\frac{\ln n}{\ln(n+1)}=b|x-a|

where |b|=b because b>0 is given. So the series converges when

b|x-a|

This means the radius of convergence is \dfrac1b.

- - -

10. By the ratio test, the series converges if

\displaystyle\lim_{n\to\infty}\left|\frac{\frac{x^{2(n+1)}}{(n+1)(\ln(n+1))^2}}{\frac{x^{2n}}{n(\ln n)^2}}\right|

The limit is

\displaystyle|x^2|

and so the radius of convergence is 1.

- - -

11. Incorrect. By the root test, the series converges for

\displaystyle\lim_{n\to\infty}\sqrt[n]{\left|\frac{(x-2)^n}{n^n}\right|}=\lim_{n\to\infty}\frac{|x-2|}n=0

which means the series converges for all x, and so the interval of convergence is (-\infty,\infty).

- - -

For 14 and 16, it'll probably be too late to edit this post by the time you see this. You can try posting the remaining problems in a new question.
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Determine formula of the nth term 2, 6, 12 20 30,42​
nalin [4]

Check the forward differences of the sequence.

If \{a_n\} = \{2,6,12,20,30,42,\ldots\}, then let \{b_n\} be the sequence of first-order differences of \{a_n\}. That is, for n ≥ 1,

b_n = a_{n+1} - a_n

so that \{b_n\} = \{4, 6, 8, 10, 12, \ldots\}.

Let \{c_n\} be the sequence of differences of \{b_n\},

c_n = b_{n+1} - b_n

and we see that this is a constant sequence, \{c_n\} = \{2, 2, 2, 2, \ldots\}. In other words, \{b_n\} is an arithmetic sequence with common difference between terms of 2. That is,

2 = b_{n+1} - b_n \implies b_{n+1} = b_n + 2

and we can solve for b_n in terms of b_1=4:

b_{n+1} = b_n + 2

b_{n+1} = (b_{n-1}+2) + 2 = b_{n-1} + 2\times2

b_{n+1} = (b_{n-2}+2) + 2\times2 = b_{n-2} + 3\times2

and so on down to

b_{n+1} = b_1 + 2n \implies b_{n+1} = 2n + 4 \implies b_n = 2(n-1)+4 = 2(n + 1)

We solve for a_n in the same way.

2(n+1) = a_{n+1} - a_n \implies a_{n+1} = a_n + 2(n + 1)

Then

a_{n+1} = (a_{n-1} + 2n) + 2(n+1) \\ ~~~~~~~= a_{n-1} + 2 ((n+1) + n)

a_{n+1} = (a_{n-2} + 2(n-1)) + 2((n+1)+n) \\ ~~~~~~~ = a_{n-2} + 2 ((n+1) + n + (n-1))

a_{n+1} = (a_{n-3} + 2(n-2)) + 2((n+1)+n+(n-1)) \\ ~~~~~~~= a_{n-3} + 2 ((n+1) + n + (n-1) + (n-2))

and so on down to

a_{n+1} = a_1 + 2 \displaystyle \sum_{k=2}^{n+1} k = 2 + 2 \times \frac{n(n+3)}2

\implies a_{n+1} = n^2 + 3n + 2 \implies \boxed{a_n = n^2 + n}

6 0
2 years ago
50 POINTS!!! In rectangle ABCD, AB = 6 cm, BC = 8 cm, and DE = DF. The area of triangle DEF is one-fourth the area of rectangle
aalyn [17]

Answer:

EF=4\sqrt{3}

Step-by-step explanation:

In rectangle ABCD, AB = 6, BC = 8, and DE = DF.

ΔDEF is one-fourth the area of rectangle ABCD.

We want to determine the length of EF.

First, we can find the area of the rectangle. Since the length AB and width BC measures 6 by 8, the area of the rectangle is:

A_{\text{rect}}=8(6)=48\text{ cm}^2

The area of the triangle is 1/4 of this. Therefore:

\displaystyle A_{\text{tri}}=\frac{1}{4}(48)=12\text{ cm}^2

The area of a triangle is half of its base times its height. The base and height of the triangle is DE and DF. Therefore:

\displaystyle 12=\frac{1}{2}(DE)(DF)

Since DE = DF:

24=DF^2

Thus:

DF=\sqrt{24}=\sqrt{4\cdot 6}=2\sqrt{6}=DE

Since ABCD is a rectangle, ∠D is a right angle. Then by the Pythagorean Theorem:

(DE)^2+(DF)^2=(EF)^2

Therefore:

(2\sqrt6)^2+(2\sqrt6)^2=EF^2

Square:

24+24=EF^2

Add:

EF^2=48

And finally, we can take the square root of both sides:

EF=\sqrt{48}=\sqrt{16\cdot 3}=4\sqrt{3}

6 0
3 years ago
Read 2 more answers
28 is what percent of 50?
nikdorinn [45]

Answer:

go to 100

56 is 56 percent of 100

28 is 56 percent of 50

Step-by-step explanation:

6 0
3 years ago
Math <br><br><br> help pls!!<br><br> solve for x and y
Elan Coil [88]

Answer:

x = 7 ,  y= - 6

Step-by-step explanation:

4x = -2 - 5y ___(1)

4x= 10 - 3y ___(2)

so, now by simultaneous equations,

10 - 3y = - 2 - 5y

2y = -12

y = -6

if y = -6

then, 4x= 10 -3(-6)

4x =28

x = 7

8 0
3 years ago
How long can the run be made?
defon

Answer:

L=15\ m

Step-by-step explanation:

we know that

The perimeter of rectangle is equal to

P=2L+2W

In this problem we have

P=42\ m

W=6\ m

substitute in the formula and solve for L

42=2L+2(6)

simplify

21=L+(6)

L=21-6=15\ m

6 0
3 years ago
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