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 .
- 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 ) as it has a finite number of terms.
- On the other hand, the second sequence (with ) 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: , where first term and common ratio.
- The term of a geometric sequence is, .
All about Geometric Series
- It is the sum of the terms of a geometric sequence.
- The formula to find the sum of '' terms of a finite geometric sequence is , provided .
- The formula to find the sum of '' terms of an infinite geometric sequence with is .
Example 1: Is 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 ) and hence is an arithmetic sequence.]
Example 2: Find the first six terms of a geometric sequence in which the first term is and the common ratio is .
Solution: It is given that the first term is and the common ratio is . We know that the of a a geometric sequence is . We can find the other terms using this formula.
Therefore, the first six terms of the given geometric sequence are and.
Example 3: Find the next term of a geometric sequence
Solution: Here, the first four terms of the geometric sequence are given. The first term is, . We can calculate the common ratio, .
Now, the next term means the fifth term i.e.
Example 4: Find the sum of the first ten terms of the given geometric sequence:
Solution: Here, the first term , common ratio . We know that, .
Example 5: Find the sum of the infinite geometric series
Solution: Here, the first term , common ratio . We know that for , .
- The common ratio of any geometric sequence is found by dividing any term of the geometric sequence by its preceding term i.e.
- term of a geometric sequence from the beginning:
- term of a finite geometric sequence (having '' terms) from the end:
- For a finite geometric sequence, the sum of the first '' terms: , where .
- For , the sum of the first '' terms of the finite geometric sequence: .
- For an infinite geometric sequence where , the sum of first '' terms: .
- For an infinite geometric sequence where , the sum cannot be computed.
- If represents a set (group) of three consecutive terms of any geometric sequence, then is called the geometric mean of the set, and its value is .
- 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.