Algebra 2 - Geometric Sequence & Series


All about Geometric Sequences

  • A geometric sequence (aka geometric progression) is a special type of sequence in which the ratio between two consecutive terms is always the same (constant).
  • The ratio is called common ratio (r).
  • Some examples of geometric sequences are given in the figure below.

  • Based on the number of terms, geometric sequences are classified into two types - finite and infinite.
  • The first sequence shown in the figure above represents a finite geometric sequence (with r=2) as it has a finite number of terms.
  • On the other hand, the second sequence (with r=12) has an infinite number of terms (indicated by dots) and hence is an example of an infinite geometric sequence.
  • In general, a geometric sequence is represented as: a1, a1r, a1r2, a1r3, a1r4, ...., where a1=first term and r=common ratio.
  • The nth term of a geometric sequence is, an=a1rn-1.


All about Geometric Series

  • It is the sum of the terms of a geometric sequence.
  • The formula to find the sum of 'n' terms of a finite geometric sequence is Sn=a1·(1-rn)(1-r), provided r1.
  • The formula to find the sum of 'n' terms of an infinite geometric sequence with |r|<1 is S=a1(1-r).

Solved Examples

Example 1: Is 5, 10, 15, 20, 25 a geometric sequence?

Solution: The condition for a sequence to be a geometric sequence is that it must have a common ratio. Since this condition is not satisfied for the given sequence, it is not a geometric sequence.

[Note: In the given sequence, the difference between any two consecutive terms is the same (common difference d=5) and hence is an arithmetic sequence.]


Example 2: Find the first six terms of a geometric sequence in which the first term is a1=3 and the common ratio is r=2.

Solution: It is given that the first term is a1=3 and the common ratio is r=2. We know that the nth of a a geometric sequence is an=a1rn-1. We can find the other terms using this formula.

Second term, a2=3·22-1=3·21=6

Third term, a3=3·23-1=3·22=12

Fourth term, a4=3·24-1=3·23=24

Fifth term, a5=3·25-1=3·24=48

Sixth term, a6=3·26-1=3·25=96

Therefore, the first six terms of the given geometric sequence are 3, 6, 12, 24, 48, and 96.


Example 3: Find the next term of a geometric sequence 5, 1, 15, 125. . .

Solution: Here, the first four terms of the geometric sequence are given. The first term is, a1=5. We can calculate the common ratio, r=15.

Now, the next term means the fifth term i.e. a5=5·155-1=5·154=5·15×5×5×5=1125


Example 4: Find the sum of the first ten terms of the given geometric sequence: 2, -6, 18, -54, . . .

Solution: Here, the first term a1=2, common ratio r=-62=-3. We know that, Sn=a1·1-rn1-r.



Example 5: Find the sum of the infinite geometric series 27, 9, 3, 1, 13, 19, . . . . 

Solution: Here, the first term a1=27, common ratio d=927=13. We know that for r<1S=a(1-r).


Cheat Sheet

  • The common ratio of any geometric sequence is found by dividing any term of the geometric sequence by its preceding term i.e. r=anan-1
  • nth term of a geometric sequence from the beginning: an=a1·rn-1
  • mth term of a finite geometric sequence (having 'n' terms) from the end: am=a1·rn-m
  • For a finite geometric sequence, the sum of the first 'n' terms: Sn=a1·(1-rn)(1-r), where r1.
  • For r=1, the sum of the first 'n' terms of the finite geometric sequence: Sn=na1.
  • For an infinite geometric sequence where |r|<1, the sum of first 'n' terms: S=a1(1-r).
  • For an infinite geometric sequence where |r|1, the sum cannot be computed.
  • If x, y, z represents a set (group) of three consecutive terms of any geometric sequence, then y is called the geometric mean of the set, and its value is y=xz.

Blunder Areas

  • When each term of a geometric sequence is multiplied by a non-zero number, the resulting sequence is also a geometric sequence with the same common ratio.
  • The new sequence obtained by taking the reciprocal of terms of a geometric sequence is also a geometric sequence.
  • If all the terms of a given geometric sequence are raised to the same power, then the resulting sequence is also a geometric sequence.