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3 years ago

What is the nth term rule of the linear sequence below?

Mathematics

1 answer:

Katena32 [7]3 years ago

6 0

Answer:

4n -13

Step-by-step explanation:

notice that the difference between the adjacent terms is 4.

nth term is given by:

nth term = first term + (n-1) × difference -n is any number

= -9 + (n-1) × 4

= -9 + 4n -4

= 4n -13

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Solve using Fourier series.

Olin [163]

With $2L=\pi$ , the Fourier series expansion of $f(x)$ is

$\displaystyle f(x)\sim\frac{a_0}2+\sum_{n\ge1}a_n\cos\dfrac{n\pi x}L+\sum_{n\ge1}b_n\sin\dfrac{n\pi x}L$
$\displaystyle f(x)\sim\frac{a_0}2+\sum_{n\ge1}a_n\cos2nx+\sum_{n\ge1}b_n\sin2nx$

where the coefficients are obtained by computing

$\displaystyle a_0=\frac1L\int_0^{2L}f(x)\,\mathrm dx$
$\displaystyle a_0=\frac2\pi\int_0^\pi f(x)\,\mathrm dx$

$\displaystyle a_n=\frac1L\int_0^{2L}f(x)\cos\dfrac{n\pi x}L\,\mathrm dx$
$\displaystyle a_n=\frac2\pi\int_0^\pi f(x)\cos2nx\,\mathrm dx$

$\displaystyle b_n=\frac1L\int_0^{2L}f(x)\sin\dfrac{n\pi x}L\,\mathrm dx$
$\displaystyle b_n=\frac2\pi\int_0^\pi f(x)\sin2nx\,\mathrm dx$

You should end up with

$a_0=0$
$a_n=0$
(both due to the fact that $f(x)$ is odd)
$b_n=\dfrac1{3n}\left(2-\cos\dfrac{2n\pi}3-\cos\dfrac{4n\pi}3\right)$

Now the problem is that this expansion does not match the given one. As a matter of fact, since $f(x)$ is odd, there is no cosine series. So I'm starting to think this question is missing some initial details.

One possibility is that you're actually supposed to use the even extension of $f(x)$ , which is to say we're actually considering the function

$\varphi(x)=\begin{cases}\frac\pi3&\text{for }|x|\le\frac\pi3\\0&\text{for }\frac\pi3$

and enforcing a period of $2L=2\pi$ . Now, you should find that

$\varphi(x)\sim\dfrac2{\sqrt3}\left(\cos x-\dfrac{\cos5x}5+\dfrac{\cos7x}7-\dfrac{\cos11x}{11}+\cdots\right)$

The value of the sum can then be verified by choosing $x=0$ , which gives

$\varphi(0)=\dfrac\pi3=\dfrac2{\sqrt3}\left(1-\dfrac15+\dfrac17-\dfrac1{11}+\cdots\right)$
$\implies\dfrac\pi{2\sqrt3}=1-\dfrac15+\dfrac17-\dfrac1{11}+\cdots$

as required.

5 0

3 years ago

Which of the following completes the rule for this translation, (x, y) → (x − 8, ___)?

almond37 [142]

Answer:

The answer would be c

Step-by-step explanation:

I did it edge yesterday

4 0

3 years ago

Read 2 more answers

This is a continuous set of points in the coordinate plane. It can be, but is not always, straight.

castortr0y [4]

Answer:

Linear. While all linear equations produce straight lines when graphed, not all linear equations produce linear functions. In order to be a linear function, a graph must be both linear (a straight line) and a function (matching each x-value to only one y-value).

5 0

3 years ago

Which expressions are equivalent to 3x+3(x+y)?

storchak [24]

Answer: 6x+3y

Step-by-step explanation:

3x+3(x+y)= 3x+3x+3y=6x+3y

6 0

3 years ago

Consuela and her three friends order 2 pizzas. Consuela cuts each pizza into 8 equal slices. She saves 2 slices of pizza for her

Akimi4 [234]

Let
x--------------> the number of slices eating by Consuela and her three friends

we know that
[total slices]=8*2=16 slices
[slices of pizza for her older brother]=2
x=16-2=14 slides
Consuela and her friends share the rest of the pizza, each eating the same number of slices---------> 14/4= 3.5 slides of pizza each (this part is not necessary, but it is shown in a didactic way)

the question is
What fraction of the 2 pizzas is left over?
[slices of pizza for her older brother]/[total slices]=2/16=1/8

the answer is (1/8)

8 0

3 years ago