Algebra - Geometric Progression (GP)

A Geometric Progression (GP) is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$). Visually, this creates a pattern of constant scaling rather than constant addition.

The Common Ratio ($r$) is the constant multiplier. It is calculated by dividing any term by its preceding term, such as $r = \frac{T_2}{T_1} = \frac{T_3}{T_2}$. If $r > 1$, the sequence grows larger (diverges), while if $0 < r < 1$, the terms get smaller (converges) towards zero.

The General Form of a GP is represented as $a, ar, ar^2, ar^3, \dots, ar^{n-1}$, where $a$ is the first term and $n$ is the number of terms. On a graph, a GP does not form a straight line like an Arithmetic Progression; instead, it forms an exponential curve that becomes steeper as $n$ increases if $r > 1$.

The General Term or $n^{th}$ term ($T_n$) allows you to find any specific position in the sequence. It is defined as $T_n = a r^{n-1}$. This formula shows that the exponent of the ratio is always one less than the position of the term.

When solving problems involving three numbers in a GP, it is often visually and algebraically simpler to assume the terms are $\frac{a}{r}, a, ar$. This is because their product is simply $a^3$, as the $r$ terms cancel out, simplifying the calculation of the middle term.

The Sum of $n$ terms ($S_n$) represents the total value of all terms in the sequence up to $n$. The formula changes slightly depending on whether $r$ is greater than or less than 1 to keep the denominator positive, but both forms are mathematically equivalent.

The Geometric Mean (GM) between two numbers $a$ and $b$ is a number $G$ such that $a, G, b$ are in GP. Algebraically, $G^2 = ab$ or $G = \sqrt{ab}$. Visually, the GM represents the side of a square that has the same area as a rectangle with sides $a$ and $b$.