3 years ago

Determine whether the sequence shown are arithmetic or geometric 1,8,15,22,29

1 answer:

5 0

1

8 <--- 1 + 7

15 <--- 8 + 7

22 <--- 15 + 7

29 <--- 22 + 7

is simply adding "7" to get the next value, so is an arithmetic sequence, whose common difference is 7.

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Take the derivative to get

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$\implies\dfrac x{(1-x)^2}=\displaystyle\sum_{k\ge0}kx^k$

Take the derivative again:

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Take the derivative one more time:

$\displaystyle\frac{(1+2x)(1-x)^3+3(x+x^2)(1-x)^2}{(1-x)^6}=\sum_{k\ge0}k^3x^{k-1}=\frac1x\sum_{k\ge0}k^3x^k$

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so we have

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

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