Mathematics: Geometric Sequence

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In a geometric sequence of numbers, the common ratio between two terms is a constant.

For the general geometric sequence a₁, a₂, a₃, ... , a_n, ... , the common ratio r = a_n / a_n-1

The general term is   a_n = a₁ r^n-1  where a_n is the nth term and r is the common ratio

Example geometric sequences are;

• 1, 2, 4, 8, 16, 32, ... , where r = 2
• 1, -2, 4, -8, 16, -32, ... , where r = -2
• 50, 5, 0.5, 0.05, ... , where r = 0.1
• 4, 2, 1, ½, ¼, ... , where r = ½

The relationship between n and a_n is nonlinear.

Problem Example

• If a₆ = -4 and a₇ = -20 then r =  -20/-4 = 5
• If a₂ = 8 and a₄ = 32 then r = 32/a₃ = a₃/8  => 256 = (a₃)²  => a₃ = 16 => r = 2
• If a₄ = 54 and a₇ = 1458 then
  => ar³ = 54, ar⁶ = 1458 => 54/r³ = 1485/r⁶ => r³ = 1485/54 = 27 => r = 3
  => 54, 162, 486, 1458, ...