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vovikov84 [41]
2 years ago
7

Write the following fraction in decimal numbers encircle the letter of your answer

Mathematics
1 answer:
Allisa [31]2 years ago
3 0

Answer:

D  and D

Step-by-step explanation:

31. 2/10

A.0/10

B.0.40

C.0.04

D. ⊂ 0.2 ⊃

32. 12/25

A.0.12

B.0.24

C.0.36

D. ⊂ 0.48 ⊃

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pantera1 [17]

This sequence has generating function

F(x)=\displaystyle\sum_{k\ge0}k^3x^k

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Recall that for |x|, we have

\displaystyle\frac1{1-x}=\sum_{k\ge0}x^k

Take the derivative to get

\displaystyle\frac1{(1-x)^2}=\sum_{k\ge0}kx^{k-1}=\frac1x\sum_{k\ge0}kx^k

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Take the derivative again:

\displaystyle\frac{(1-x)^2+2x(1-x)}{(1-x)^4}=\sum_{k\ge0}k^2x^{k-1}=\frac1x\sum_{k\ge0}k^2x^k

\implies\displaystyle\frac{x+x^2}{(1-x)^3}=\sum_{k\ge0}k^2x^k

Take the derivative one more time:

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\implies\displaystyle\frac{x+4x^3+x^3}{(1-x)^4}=\sum_{k\ge0}k^3x^k

so we have

\boxed{F(x)=\dfrac{x+4x^3+x^3}{(1-x)^4}}

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