25.3 Geometric sequences
When you count by 5, 6, 7, 8, 9 and onwards you would realize that you’re actually counting an ordered list of numbers. Ordered list of numbers are otherwise referred to as sequences. Consider we’re given the sequence 11, 9, 7, 5, 3, 1… If you would see, there are three dots ending the list of numbers. This indicates that this sequence is infinite and it can go on as long it could.
There are two kinds of sequences, infinite, and finite. We already saw an example of infinitesequence in the first paragraph and from there we know that they can go on and on. For the finite sequence they can just be limited with a few terms. Terms or Elements are the numbers in a sequence. The sequence has a common difference and a common ratio. This would help establish the relationship between the successive terms in the sequence. When we determine the sum of a sequence then we are able to find the series.
Apart from looking at the regular numbers like 1,2,3,4,5 and so on, we will be able to study other expressions which might be consisted of a coefficient, a variable and an exponent and other expressions such as factorials. In this chapter, we will also learn how to use mathematical induction to prove the sum of the series.
Geometric sequences
Basic concepts:
 Arithmetic sequences
Lessons
Notes:
• geometric sequence: a sequence with a common ratio between successive terms.
• the nth term, ${t_n}$ ,of a geometric sequence:
${t_n}\; = \;{t_1} \cdot {r^{n  1}}$
where, ${t_n}$ : nth term
${t_1}$ : first term
r : common ratio

1.
Geometric sequence formula
Consider the geometric sequence: 2, 6, 18, 54, … .