3 szkoły ponadpodstawowej

III liceum

III technikum

Matematyka z plusem 3. Zakres podstawowy i rozszerzony

a)   bn​=−5an​   Ciąg  (an​)  jest rosnący, więc: an+1​−an​&gt;0   Obliczamy różnicę bn+1​−bn​,  aby określić monotoniczność ciągu  (bn​).

a) Wyrazy danego ciągu zapiszemy w innej postaci. 21​, 42​, 83​, 164​, 325​, 646​, 1287​, 2168​, …   211​, 222​, 233​, 244​, 255​, 266​, 277​, 288​, …

a) Wyrazy danego ciągu zapiszemy w innej postaci. 3, 32, (32)2, ((32)2)2,  …   Wzór rekurencyjny:

Najpierw podamy przybliżenie liczby  2<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
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M834 80h400000v40h-400000z'/></svg>​  z dokładnością do części tysięcznych. 2<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
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l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg>​=1,4142135…≈1,414   Wyznaczymy kilka początkowych wyrazów ciągu  (an​).

a) Wyznaczmy pierwszy wyraz ciągu  (an​).   a1​=12−1=1−1=0   Wyznaczmy teraz wyraz  an+1​.

Wyznaczymy kilka kolejnych wyrazów ciągu  (an​).   a1​=7   a2​=14

a) Obliczamy różnicę an+1​−an​,  aby określić monotoniczność ciągu.  an+1​−an​=8−n+1<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
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H400000v40H845.2724
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c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg>​−(8−n<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
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c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg>​)=8−n+1<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
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H400000v40H845.2724
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M834 80h400000v40h-400000z'/></svg>​−8+n<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
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c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
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M834 80h400000v40h-400000z'/></svg>​=n<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
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c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
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H400000v40H845.2724
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M834 80h400000v40h-400000z'/></svg>​−n+1<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
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M834 80h400000v40h-400000z'/></svg>​

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