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Geometric sequence

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A geometric sequence (or geometric progression) is a (finite or infinite) sequence of (real or complex) numbers such that the quotient (or ratio) of consecutive elements is the same for every pair.

In finance, compound interest generates a geometric sequence.

Examples

Examples for geometric sequences are

• $3,6,12,24,48,96$ (finite, length 6: 6 elements, quotient 2)

• $1,-2,4,-8$ (finite, length 4: 4 elements, quotient −2)

• $8,4,2,1,{1 \over 2},{1 \over 4},{1 \over 8},\dots {1 \over 2^{n-4}},\dots$ (infinite, quotient $1 \over 2$)

• $2,2,2,2,\dots$ (infinite, quotient 1)

• $-2,2,-2,2,\dots ,(-1)^{n}\cdot 2,\dots$ (infinite, quotient −1)

• ${1 \over 2},1,2,4,\dots ,2^{n-2},\dots$ (infinite, quotient 2)

• $1,0,0,0,\dots \$ (infinite, quotient 0) (See General form below)

Application in finance

The computation of compound interest leads to a geometric series:

When an initial amount A is deposited at an interest rate of p percent per time period then the value A_n of the deposit after n time-periods is given by

$A_{n}=A\left(1+{p \over 100}\right)^{n}$

i.e., the values A=A₀, A₁, A₂, A₃, ... form a geometric sequence with quotient q = 1+(p/100).

.... (read more)

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