幾何級数
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*幾何級数:geometric Progression [#r7f85dff]
等比数列(とうひすうれつ、または幾何数列(きかすうれつ)...
-In mathematics, a geometric progression, also known as a...
-The behaviour of a geometric sequence depends on the val...
If the common ratio is:
--Positive, the terms will all be the same sign as the in...
--Negative, the terms will alternate between positive and...
--Greater than 1, there will be exponential growth toward...
1, the progression is a constant sequence.
--Between −1 and 1 but not zero, there will be exponentia...
−1, the progression is an alternating sequence (see alter...
--Less than −1, for the absolute values there is exponent...
*等比数列の一般式 [#ce7078c9]
a1 = a
an+1 = r ・an (n≧1)
*等比数列 [#s7115ee8]
Thus, the general form of a geometric sequence is
a,ar,ar^2,ar^3,・・・・・・
and that of a geometric series is
a+ar+ar^2+ar^3+ar^4・・・・・・
where r ≠ 0 is the common ratio and a is a scale factor, ...
*幾何級数:等比級数 [#v5a49c12]
等比級数は等比数列の項の総和のことをいい、初項から第n+1項...
#ref(geometricprogression.png)
「証明]
#ref(geometricseriese.JPG)
*成長の表現:螺旋状の成長 [#j9f7e5af]
「ものすごい勢いで増える」ことを、「幾何級数的に増える」...
-幾何級数的増加と線型的増加
--アルキメデスの螺旋は等差級数的に増加
--等角螺旋(対数螺旋)は、幾何級数的に増加
#ref(geometricprogression2.JPG)
*成長の限界 The Limits to Growth [#h406b6f5]
The Limits to Growth is a 1972 book modeling the conseque...
One key idea that The Limits to Growth discusses is that ...
#ref(limitofgrowth.png)
(note that the book rounded off numbers).
-In general, the formula for calculating the amount of ti...
#ref(limitofgrowth2.png)
where:
y = years left;
g = 1.026 (2.6% annual consumption growth);
R = reserve;
C = (annual) consumption.
*資源が消費で枯渇する年数の計算 [#df5def3d]
Annual onsumption growth rate and the years left.
Chromium: 2.6% 95
Gold 4.1% 9
Iron 1.8% 93
Petroleum 3.9% 20
Limits to Growth has had a huge impact on how we still th...
終了行:
*幾何級数:geometric Progression [#r7f85dff]
等比数列(とうひすうれつ、または幾何数列(きかすうれつ)...
-In mathematics, a geometric progression, also known as a...
-The behaviour of a geometric sequence depends on the val...
If the common ratio is:
--Positive, the terms will all be the same sign as the in...
--Negative, the terms will alternate between positive and...
--Greater than 1, there will be exponential growth toward...
1, the progression is a constant sequence.
--Between −1 and 1 but not zero, there will be exponentia...
−1, the progression is an alternating sequence (see alter...
--Less than −1, for the absolute values there is exponent...
*等比数列の一般式 [#ce7078c9]
a1 = a
an+1 = r ・an (n≧1)
*等比数列 [#s7115ee8]
Thus, the general form of a geometric sequence is
a,ar,ar^2,ar^3,・・・・・・
and that of a geometric series is
a+ar+ar^2+ar^3+ar^4・・・・・・
where r ≠ 0 is the common ratio and a is a scale factor, ...
*幾何級数:等比級数 [#v5a49c12]
等比級数は等比数列の項の総和のことをいい、初項から第n+1項...
#ref(geometricprogression.png)
「証明]
#ref(geometricseriese.JPG)
*成長の表現:螺旋状の成長 [#j9f7e5af]
「ものすごい勢いで増える」ことを、「幾何級数的に増える」...
-幾何級数的増加と線型的増加
--アルキメデスの螺旋は等差級数的に増加
--等角螺旋(対数螺旋)は、幾何級数的に増加
#ref(geometricprogression2.JPG)
*成長の限界 The Limits to Growth [#h406b6f5]
The Limits to Growth is a 1972 book modeling the conseque...
One key idea that The Limits to Growth discusses is that ...
#ref(limitofgrowth.png)
(note that the book rounded off numbers).
-In general, the formula for calculating the amount of ti...
#ref(limitofgrowth2.png)
where:
y = years left;
g = 1.026 (2.6% annual consumption growth);
R = reserve;
C = (annual) consumption.
*資源が消費で枯渇する年数の計算 [#df5def3d]
Annual onsumption growth rate and the years left.
Chromium: 2.6% 95
Gold 4.1% 9
Iron 1.8% 93
Petroleum 3.9% 20
Limits to Growth has had a huge impact on how we still th...
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