Understanding Arithmetic, Geometric, and Harmonic Progressions

A progression is a sequence of numbers such that the relationship between consecutive terms is described by a specific arithmetic, geometric, or reciprocal pattern. The principal progressions in mathematics are arithmetic progression (AP), geometric progression (GP), and harmonic progression (HP). Each progression obeys a precise formula for its general term and sum, which is foundational in algebraic problem-solving.

Formal Construction of Arithmetic Progression (AP)

An arithmetic progression is a sequence of the form $a_1, a_2, a_3, \ldots, a_n, \ldots$ where a fixed real number $d$ exists such that $a_{k+1} - a_k = d$ for every $k \in \mathbb{N}$. The quantity $d$ is called the common difference.

The general or $n$-th term of an arithmetic progression is given by $a_n = a_1 + (n-1)d$.

The sum of the first $n$ terms of an arithmetic progression, denoted $S_n$, is derived as follows. By definition,

$S_n = a_1 + a_2 + a_3 + \cdots + a_n$

Explicitly writing and then reversing the order,

$S_n = a_1 + [a_1 + d] + [a_1 + 2d] + \cdots + [a_1 + (n-1)d]$

$S_n = [a_1 + (n-1)d] + [a_1 + (n-2)d] + \cdots + a_1$

Adding these two representations termwise,

$2S_n = [a_1 + a_1 + (n-1)d] + [a_1 + d + a_1 + (n-2)d] + \cdots + [a_1 + (n-1)d + a_1]$

Each pair adds to $2a_1 + (n-1)d$. There are $n$ such pairs, thus,

$2S_n = n[2a_1 + (n-1)d]$

Dividing both sides by $2$,

$S_n = \dfrac{n}{2}[2a_1 + (n-1)d]$

Alternatively, since $a_n = a_1 + (n-1)d$, it follows that $S_n = \dfrac{n}{2}(a_1 + a_n)$.

Arithmetic Geometric And Harmonic Progression

Explicit Construction of Geometric Progression (GP)

A geometric progression is a sequence $g_1, g_2, g_3, \ldots, g_n, \ldots$ in which the ratio of any term to its predecessor is constant for all $k \in \mathbb{N}$. This constant $r$ is the common ratio, and $g_{k+1} = r\,g_k$.

The general or $n$-th term of a geometric progression is $g_n = g_1\,r^{n-1}$.

The sum of the first $n$ terms, $S_n$, for $r \neq 1$ is established by direct computation. Start with

$S_n = g_1 + g_1 r + g_1 r^2 + \cdots + g_1 r^{n-1}$

Multiply both sides by $r$:

$rS_n = g_1 r + g_1 r^2 + \cdots + g_1 r^n$

Subtract the second equation from the first:

$S_n - rS_n = [g_1 + g_1 r + \cdots + g_1 r^{n-1}] - [g_1 r + g_1 r^2 + \cdots + g_1 r^n]$

$S_n - rS_n = g_1 - g_1 r^n$

$S_n (1 - r) = g_1 (1 - r^n)$

Thus, for $r \neq 1$:

$S_n = \dfrac{g_1(1 - r^n)}{1 - r}$

If $r > 1$, the numerator may be written as $r^n - 1$ and denominator as $r - 1$, yielding $S_n = g_1 \left(\dfrac{r^n - 1}{r - 1}\right)$.

Geometric Mean Overview

Mathematical Characterisation of Harmonic Progression (HP)

A sequence $h_1, h_2, h_3, \ldots$ is in harmonic progression if the sequence of reciprocals $1/h_1, 1/h_2, 1/h_3, \ldots$ forms an arithmetic progression. Thus, if $a_k = 1/h_k$, then $a_k$ is in arithmetic progression.

If $a, a + d, a + 2d, \ldots$ is the underlying AP, then $h_k = \dfrac{1}{a + (k-1)d}$ for all $k \in \mathbb{N}$.

The $n$-th term of the harmonic progression is $h_n = \dfrac{1}{a + (n-1)d}$, with $a \neq 0$ and $a + (n-1)d \neq 0$ for $n \geq 1$.

Harmonic Progression Explained

Sum of Finite Harmonic Progression: Analytical Limits

The sum of the first $n$ terms of HP, $S_n$, does not admit a direct closed form as for AP and GP. If $h_k = \dfrac{1}{a + (k-1)d}$, then

$S_n = \sum_{k=1}^{n} \dfrac{1}{a + (k-1)d}$

This expression is typically evaluated individually using properties of telescoping sums or, for small values of $n$, direct calculation. For large $n$ and $d\neq 0$, advanced analytic techniques or integral approximations are required.

Example Evaluations of Each Progression

AP Example: Consider the progression $7, 13, 19, 25, 31$. The first term is $a_1 = 7$, the common difference is $d = 6$. The $n$-th term is $a_n = 7 + (n-1) \times 6$. For $n = 5$, $a_5 = 7 + 4 \times 6 = 7 + 24 = 31$.

The sum of $5$ terms: $S_5 = \dfrac{5}{2}(7 + 31) = \dfrac{5}{2} \times 38 = 5 \times 19 = 95$.

GP Example: For the progression $5, 10, 20, 40$, the first term is $g_1 = 5$, the common ratio is $r = 2$. The $n$-th term is $g_n = 5 \times 2^{n-1}$. For $n=4$, $g_4 = 5 \times 2^{3} = 5 \times 8 = 40$.

The sum of $4$ terms: $S_4 = 5\,\dfrac{2^4 - 1}{2 - 1} = 5\,\dfrac{16-1}{1} = 5 \times 15 = 75$.

HP Example: For the sequence $\dfrac{1}{2}, \dfrac{1}{5}, \dfrac{1}{8}, \dfrac{1}{11}$, $a=2, d=3$. The $n$-th term is $h_n = \dfrac{1}{2 + (n-1)3}$. For $n=4$, $h_4 = \dfrac{1}{2+9} = \dfrac{1}{11}$.

Arithmetic Progression Practice Paper

Relation and Inequality among Arithmetic, Geometric, and Harmonic Progression Means

For any positive real numbers $x_1, x_2, \ldots, x_n$, the arithmetic mean is $A = \dfrac{1}{n}\sum_{k=1}^n x_k$, the geometric mean is $G = \left(\prod_{k=1}^n x_k\right)^{1/n}$, and the harmonic mean is $H = \dfrac{n}{\sum_{k=1}^n \frac{1}{x_k}}$.

The classical inequality holds: $A \geq G \geq H$, with equality if and only if all $x_k$ are equal.

Difference Between Means

Analysis of Convergence in Infinite Geometric Progressions

Consider the infinite geometric series $\sum_{k=0}^{\infty} g_1 r^k$. This series converges if and only if $|r| < 1$. In that case, the sum is $S_\infty = \dfrac{g_1}{1 - r}$, derived from the finite sum formula as $n \to \infty$.

When $|r| \geq 1$, the infinite geometric series does not approach a finite limit and is deemed divergent.

Distinction Between Arithmetic, Geometric, and Harmonic Progressions

The defining characteristic of an arithmetic progression is fixed additive change, whereas a geometric progression involves constant multiplicative change and a harmonic progression involves reciprocals of an arithmetic progression. In an AP, term increments are linear; in a GP, they are exponential; in an HP, term values decrease harmonically and are not generally equidistant.

The infinite arithmetic progression does not converge if the common difference $d \neq 0$, whereas an infinite geometric progression converges exactly when $|r| < 1$. Harmonic progression convergence is governed by the divergence or convergence of the underlying AP’s reciprocals.