Sequences and Series - Sequences and Series

A sequence is a collection of numbers arranged in a specific order, such as $a_1, a_2, a_3, ..., a_n$. A series is the expression obtained by adding the terms of a sequence, often represented using the sigma notation $\sum a_i$. Visually, a sequence can be represented as discrete dots on a coordinate plane where the x-axis represents the term number and the y-axis represents the term value.

An Arithmetic Progression (AP) is a sequence in which the difference between any two consecutive terms is a constant called the common difference $d$. Visually, if you plot the terms of an AP against their position $n$, the points will lie on a straight line with a slope equal to $d$. If $d > 0$, the line slopes upward; if $d < 0$, it slopes downward.

The Arithmetic Mean (AM) of two numbers $a$ and $b$ is the value $A = \frac{a+b}{2}$. Geometrically, the AM represents the midpoint between two points on a number line. If $n$ numbers are inserted between $a$ and $b$ such that the resulting sequence is an AP, these are called $n$ arithmetic means.

A Geometric Progression (GP) is a sequence where each term is obtained by multiplying the preceding term by a constant non-zero number $r$, known as the common ratio. Visually, the terms of a GP follow an exponential curve. If $r > 1$, the sequence grows rapidly; if $0 < r < 1$, the sequence decays and the points approach the x-axis asymptotically.

The Geometric Mean (GM) for two positive numbers $a$ and $b$ is given by $G = \sqrt{ab}$. In a geometric context, if a semicircle is drawn with a diameter equal to $a + b$, the length of the perpendicular segment from the meeting point of $a$ and $b$ to the circle's boundary is the GM.

The relationship between AM and GM states that for any two positive real numbers, the Arithmetic Mean is always greater than or equal to the Geometric Mean ($A \ge G$). This inequality is a fundamental property used to find minimum or maximum values in various mathematical problems.

The sum of the first $n$ terms of a sequence is denoted by $S_n$. For an AP, this sum represents the total accumulated value which grows quadratically with $n$. For a GP, the sum depends on whether the ratio $r$ is greater or less than 1, representing how quickly the total magnitude of the series is scaling.