Geometric Progressions (GP)
Understanding GP will help contrast the properties and applications of different types of sequences.
Geometric Progressions (GP) are sequences where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
The general form can be expressed as a, ar, ar², ar³, ..., where 'a' is the first term, and 'r' is the common ratio.
GPs have unique properties, such as the nth term being given by a * r^(n-1) and the sum of the first n terms calculated using the formula S_n = a(1 - r^n)/(1 - r) for r ≠ 1.
They are widely used in finance for calculating compound interest and in population studies to model growth rates.
In competitive exams, understanding GPs can aid in solving problems related to sequences and series effectively, which is crucial for both Prelims and Mains.
This topic also bridges into the study of exponential functions, enhancing mathematical fluency.