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Oksanka [162]
3 years ago
9

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:

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Step-by-step explanation:

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4 0
3 years ago
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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]

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Step-by-step explanation:

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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 <span>three friends

we know that
[total slices]=8*2=16 slices
</span><span>[slices of pizza for her older brother]=2
x=16-2=14 slides
</span><span>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  (</span><span>this part is not necessary, but it is shown in a didactic way)
</span><span>
the question is 
</span><span>What fraction of the 2 pizzas is left over?
</span>[slices of pizza for her older brother]/[total slices]=2/16=1/8
<span>
the answer is (1/8)

</span>
8 0
3 years ago
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