Sequences and Series - Geometric Progression (G.P.)

A Geometric Progression (G.P.) is a sequence where each term after the first is obtained by multiplying the preceding term by a fixed, non-zero number called the common ratio ($r$). Visually, if you plot the terms of a G.P. on a graph where the x-axis is the position $n$ and the y-axis is the value $a_n$, the points will lie on an exponential curve rather than a straight line.

The common ratio ($r$) is the constant factor found by dividing any term by its immediate predecessor, i.e., $r = \frac{a_2}{a_1} = \frac{a_3}{a_2}$. If $r > 1$, the sequence shows exponential growth, moving rapidly away from the x-axis. If $0 < r < 1$, the sequence shows exponential decay, gradually flattening and approaching the x-axis as $n$ increases.

The general term or $n^{th}$ term of a G.P. is denoted by $a_n$. It allows us to find any specific term in the sequence without listing all previous terms. If the terms alternate in sign (e.g., $2, -4, 8, -16$), the common ratio $r$ is negative, and the graph of the sequence would visually oscillate above and below the x-axis.

The sum of the first $n$ terms ($S_n$) represents the total value of the series up to that point. The formula used depends on whether $|r|$ is greater than or less than $1$ to maintain positive denominators for ease of calculation, though both versions are mathematically equivalent.

An infinite geometric series has a finite sum only if the absolute value of the common ratio is less than $1$ ($|r| < 1$). Visually, this means the terms become smaller and smaller, effectively 'vanishing' as $n$ approaches infinity, allowing the total sum to converge to a specific horizontal limit.

The Geometric Mean (G.M.) between two positive numbers $a$ and $b$ is a number $G$ such that $a, G, b$ form a G.P. Geometrically, if $a$ and $b$ are the sides of a rectangle, the G.M. is the side of a square with the same area. The relationship is expressed as $G = \sqrt{ab}$.

To insert $n$ geometric means $G_1, G_2, \dots, G_n$ between two numbers $a$ and $b$, we create a G.P. with $n+2$ terms where $a$ is the first term and $b$ is the $(n+2)^{th}$ term. This is useful for interpolating values that follow a multiplicative trend between two known endpoints.