幾何級数

幾何級数：geometric Progression †

等比数列（とうひすうれつ、または幾何数列（きかすうれつ）、英: geometric progression, geometric sequence）は、数列で、隣り合う二項の比が項番号によらず一定であるようなものである。その比のことを公比（こうひ、英: common ratio）といい、記号 r で表す。

• In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. The sum of the terms of a geometric progression is known as a geometric series.

• The behaviour of a geometric sequence depends on the value of the common ratio. If the common ratio is:

• Positive, the terms will all be the same sign as the initial term.
• Negative, the terms will alternate between positive and negative.
• Greater than 1, there will be exponential growth towards positive infinity. 1, the progression is a constant sequence.

• Between −1 and 1 but not zero, there will be exponential decay towards zero. −1, the progression is an alternating sequence (see alternating series)
• Less than −1, for the absolute values there is exponential growth towards infinity.

等比数列の一般式 †

a1 = a
an+1 = r ・an （n≧1）

等比数列 †

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, equal to the sequence's start value.

幾何級数：等比級数 †

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項までの和は以下の式で定義される。

「証明]

成長の表現：螺旋状の成長 †

「ものすごい勢いで増える」ことを、「幾何級数的に増える」と表現する

• 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
    

成長の限界 The Limits to Growth †

The Limits to Growth is a 1972 book modeling the consequences of a rapidly growing world population and finite resource supplies, commissioned by the Club of Rome.

One key idea that The Limits to Growth discusses is that if the rate of resource use is increasing, the amount of reserves cannot be calculated by simply taking the current known reserves and dividing by the current yearly usage, as is typically done to obtain a static index. For example, in 1972, the amount of chromium reserves was 775 million metric tons, of which 1.85 million metric tons were mined annually (see exponential growth). The static index is 775 / 1.85 = 418 years, but the rate of chromium consumption was growing at 2.6% annually (Limits to Growth, pp 54–71). If instead of assuming a constant rate of usage, the assumption of a constant rate of growth of 2.6% annually is made, the resource will instead last

(note that the book rounded off numbers).

• In general, the formula for calculating the amount of time left for a resource with constant consumption growth is :
  
  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 †

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 think about environmental issues.