Geometric and other sequences: rules and uses

Algebra • Sequences

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Given sequence 8, 4, 2, 1, what is r and u_n with n starting at 1?

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r = 1/2 and u_n = 8·(1/2)^{n-1}.

Key concepts

What you'll likely be quizzed about

Definition of a geometric progression

A geometric progression is a sequence in which each term after the first is found by multiplying the previous term by a constant r called the common ratio. The common ratio is positive in the specified scope, and may be rational or a surd. Because of constant multiplication, terms change by a fixed factor rather than a fixed amount.

General term and indexing conventions

Two common forms for the nth term exist: a·r^{n-1} when the first term is a and n starts at 1, and r^n when the sequence is specified from n = 0 or when a = 1. Clear identification of the chosen index is required before writing the formula. Misalignment of indexing causes errors in the term number and in substitution into formulae.

Properties of the common ratio r

If r > 1 then terms increase exponentially; if 0 < r < 1 then terms decrease towards zero. If r = 1 then the sequence is constant. A surd common ratio (for example √2) produces irrational terms when multiplied by rational starting values, but the same multiplicative rules apply. The sign of r affects alternation, but the specified scope restricts r to positive values, preventing sign alternation.

Recognising sequences of the form r^n

Sequences of the form r^n have first term r^0 = 1 if indexing begins at n = 0, or r^1 = r if indexing begins at n = 1. Recognition depends on checking successive ratios between terms: a constant ratio indicates geometric form. If each consecutive term is the previous term multiplied by the same r, the sequence fits r^n or a·r^n after adjustment for indexing.

Other common sequence types

Arithmetic sequences follow u_{n+1} = u_n + d and grow linearly; quadratic sequences follow polynomial rules like n^2 and show second differences constant; recurrence relations define later terms from earlier terms, for example u_{n+1} = k·u_n yields a geometric sequence when k is constant. Identification of differences or ratios helps classify an unknown sequence.

Limitations and domain

The specified scope restricts r to positive rational numbers or surds and n to integers, which ensures monotonic behaviour (no alternating signs) and predictable limits. Sequences that involve negative ratios, complex numbers, or non-integer indices fall outside the specified simple geometric scope and require extended methods.

Key notes

Important points to keep in mind

Always check whether indexing starts at n = 0 or n = 1 before writing the general term.

Use consecutive-term ratios to test for a geometric sequence.

Write u_n = a·r^{n-1} when the first term is given and indexing starts at 1.

Apply r = (u_m/u_k)^{1/(m-k)} to find the common ratio from two known terms.

If 0 < r < 1 then terms decrease towards zero; if r > 1 then terms grow exponentially.

Surd values of r are valid; expect irrational terms when multiplying rational starts by surds.

Distinguish geometric from arithmetic sequences by comparing ratios and differences.

Maintain consistent indexing when converting between r^n and a·r^{n-1} forms.