Geometric sequences

Geometric sequences

A geometric sequence, also called geometric progression, is a number sequence with a common ratio between successive terms. A term in a geometric sequence can be found by multiplying the previous one by a non-zero and fixed number (a common ratio).
Basic Concepts: Arithmetic sequences
Related Concepts: Pascal's triangle, Binomial theorem, Introduction to sequences, Monotonic and bounded sequences

Lessons

• geometric sequence: a sequence with a common ratio between successive terms.
• the nth term, tn{t_n} ,of a geometric sequence:
tn=t1⋅rn−1{t_n}\; = \;{t_1} \cdot {r^{n - 1}}
where, tn{t_n} : nth term
t1{t_1} : first term
r : common ratio
  • 1.
    Geometric sequence formula
    Consider the geometric sequence: 2, 6, 18, 54, … .
    a)
    Identify the common ratio.

    b)
    Determine the sixth term of the sequence.

    c)
    Which term in the sequence has a value of 39366?

  • 2.
    Determine t1,r,tnt_1,r,t_n for the sequences in which two terms are given:
    t3=18 t_3=18, t6=486t_6=486
  • 3.
    Three consecutive terms of a geometric sequence are written in the form
    5(x+2),8−x,x−2 5(x+2),8-x,x-2
    Find the common ratio and the possible value of each of the three terms.
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39.
Sequences and Series
39.1
Arithmetic sequences
39.2
Arithmetic series
39.3
Geometric sequences
39.4
Geometric series
39.5
Infinite geometric series
39.6
Sigma notation
39.7
Arithmetic mean vs. Geometric mean
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