Question

Can someone explain what geometric sequences are?

Answer 1

Explanation:

A sequence is a list of numbers.

A geometric sequence is a list of numbers such that the ratio of each number to the one before it is the same. The common ratio can be any non-zero value.

Examples

• 1, 2, 4, 8, ... common ratio is 2
• 27, 9, 3, 1, ... common ratio is 1/3
• 6, -24, 96, -384, ... common ratio is -4

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General Term

Terms of a sequence are numbered starting with 1. We sometimes use the symbol a(n) or an to refer to the n-th term. The general term of a geometric sequence, a(n), can be described by the formula ...

  a(n) = a(1)×r^(n-1) . . . . . n-th term of a geometric sequence

where a(1) is the first term, and r is the common ratio. The above example sequences have the formulas ...

• a(n) = 2^(n -1)
• a(n) = 27×(1/3)^(n -1)
• a(n) = 6×(-4)^(n -1)

You can see that these formulas are exponential in nature.

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Sum of Terms

Another useful formula for geometric sequences is the formula for the sum of n terms.

  S(n) = a(1)×(r^n -1)/(r -1) . . . . . sum of n terms of a geometric sequence

When |r| < 1, the sum converges as n approaches infinity. The infinite sum is ...

  S = a(1)/(1-r)