3.1 Arithmetic, Geometric & Harmonic Progressions (AP, GP, HP)

Arithmetic, Geometric & Harmonic Progressions (AP, GP, HP)

Sequences and series form a primary pillar of the mathematics component of the Pakistan Engineering Colleges Admission Test (ECAT). Engineering mathematics demands fluency in identifying structural patterns, evaluating finite and infinite summations, inserting statistical and algebraic means, and decomposing higher-order polynomial series under strict time limits.

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1. Mathematical Sequences vs. Series

A sequence is an ordered function $f: \mathbb{N} \to \mathbb{R}$ whose domain is the set of natural numbers. The values $f(1), f(2), f(3), \dots, f(n)$ are denoted as terms $a_1, a_2, a_3, \dots, a_n$. A series represents the formal indicated sum of the terms of a sequence:

$S_n = \sum_{k=1}^{n} a_k = a_1 + a_2 + a_3 + \dots + a_n$

If the sequence contains a finite count of terms, the series is finite. If the terms continue indefinitely ($n \to \infty$), the series is infinite, and its value is governed by the existence of the limit of partial sums $\lim_{n \to \infty} S_n$.

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2. Arithmetic Progressions (AP)

An Arithmetic Progression (AP), or arithmetic sequence, is a sequence in which each term after the first is generated by adding a constant real quantity called the common difference ($d$) to the preceding term:

$d = a_k - a_{k-1} \quad \text{for all } k \ge 2$

General Term ($n$-th Term)

Starting with first term $a_1$, successive terms evaluate to $a_2 = a_1 + d$, $a_3 = a_1 + 2d$, and by induction:

$a_n = a_1 + (n-1)d$

For any two arbitrary indices $m$ and $n$, the relation generalizes to:

$a_n = a_m + (n-m)d \implies d = \frac{a_n - a_m}{n-m}$

Sum of the First $n$ Terms ($S_n$)

The sum of an arithmetic progression can be expressed in two standard equivalent forms:

$S_n = \frac{n}{2}\left[2a_1 + (n-1)d\right] = \frac{n}{2}(a_1 + a_n)$

Test-Day Shortcut: Quadratic Character of $S_n$

The sum $S_n$ of any AP is always a quadratic function in $n$ with no constant term:

$S_n = A n^2 + B n$

Expanding the standard summation formula yields:

$S_n = \left(\frac{d}{2}\right)n^2 + \left(a_1 - \frac{d}{2}\right)n$

This gives two rapid mental calculation shortcuts:

1. Common Difference: $d = 2A$ (twice the coefficient of $n^2$).
2. First Term: $a_1 = S_1 = A + B$.
3. $n$-th Term Expression: $a_n = S_n - S_{n-1} = 2An + (B - A)$.

Example: If $S_n = 4n^2 - 3n$, the common difference is instantly $d = 2(4) = 8$, the first term is $a_1 = 4 - 3 = 1$, and $a_n = 8n - 7$.

Arithmetic Means (AM)

A single Arithmetic Mean ($A$) between two numbers $a$ and $b$ is the number that places $a, A, b$ into an AP:

$A - a = b - A \implies A = \frac{a+b}{2}$

To insert $n$ arithmetic means ($A_1, A_2, \dots, A_n$) between $a$ and $b$, the total sequence $a, A_1, A_2, \dots, A_n, b$ comprises $n+2$ terms where $a_1 = a$ and $a_{n+2} = b$:

$b = a + (n+2-1)d = a + (n+1)d \implies d = \frac{b-a}{n+1}$

The $k$-th inserted mean is:

$A_k = a + k d = a + k\left(\frac{b-a}{n+1}\right)$

Sum of $n$ Inserted Arithmetic Means: The sum of all $n$ arithmetic means inserted between $a$ and $b$ equals $n$ times the single arithmetic mean between them:

$\sum_{k=1}^n A_k = n \times \left(\frac{a+b}{2}\right) = n A$

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3. Geometric Progressions (GP)

A Geometric Progression (GP) is a sequence in which each term after the first is obtained by multiplying the preceding term by a non-zero constant called the common ratio ($r$):

$r = \frac{a_k}{a_{k-1}} \quad \text{for all } k \ge 2, \quad r \ne 0$

General Term ($n$-th Term)

Given the first term $a_1$ and common ratio $r$:

$a_n = a_1 r^{n-1}$

For any two terms at positions $m$ and $n$:

$\frac{a_n}{a_m} = r^{n-m} \implies r = \left(\frac{a_n}{a_m}\right)^{\frac{1}{n-m}}$

Sum of the First $n$ Terms ($S_n$)

Multiplying $S_n = a_1 + a_1 r + \dots + a_1 r^{n-1}$ by $r$ and subtracting yields:

$S_n = \frac{a_1(1-r^n)}{1-r} = \frac{a_1(r^n-1)}{r-1} \quad (r \ne 1)$

When $r = 1$, $S_n = n a_1$.

Infinite Geometric Series ($S_\infty$)

An infinite geometric series converges to a finite sum if and only if $|r| < 1$ (i.e., $-1 < r < 1$). Under this condition, $\lim_{n \to \infty} r^n = 0$, giving:

$S_\infty = \lim_{n \to \infty} \frac{a_1(1-r^n)}{1-r} = \frac{a_1}{1-r} \quad (|r| < 1)$

If $|r| \ge 1$ and $a_1 \ne 0$, the series diverges and no finite sum exists.

Application to Recurring Decimals: To convert $0.\overline{18} = 0.181818\dots$ to a fraction, express it as an infinite GP with $a_1 = \frac{18}{100}$ and $r = \frac{1}{100}$:

$S_\infty = \frac{\frac{18}{100}}{1 - \frac{1}{100}} = \frac{\frac{18}{100}}{\frac{99}{100}} = \frac{18}{99} = \frac{2}{11}$

Geometric Means (GM)

A single Geometric Mean ($G$) between two non-zero numbers $a$ and $b$ satisfies $\frac{G}{a} = \frac{b}{G}$, meaning:

$G = \pm \sqrt{ab}$

Sign convention: If $a$ and $b$ are both positive, $G = +\sqrt{ab}$. If $a$ and $b$ are both negative, $G = -\sqrt{ab}$. If $a$ and $b$ have opposite signs, their geometric mean is imaginary.

To insert $n$ geometric means ($G_1, G_2, \dots, G_n$) between $a$ and $b$, the full sequence contains $n+2$ terms where $a_1 = a$ and $a_{n+2} = b$:

$b = a r^{n+1} \implies r = \left(\frac{b}{a}\right)^{\frac{1}{n+1}}$

The $k$-th geometric mean is $G_k = a r^k = a \left(\frac{b}{a}\right)^{\frac{k}{n+1}}$.

Product of $n$ Inserted Geometric Means: The product of all $n$ geometric means inserted between $a$ and $b$ equals the $n$-th power of their single geometric mean:

$\prod_{k=1}^n G_k = G_1 \cdot G_2 \dots G_n = (\sqrt{ab})^n = G^n$

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4. Harmonic Progressions (HP)

A sequence $h_1, h_2, h_3, \dots, h_n$ is a Harmonic Progression (HP) if and only if the reciprocals of its terms form an Arithmetic Progression:

$\frac{1}{h_1}, \frac{1}{h_2}, \frac{1}{h_3}, \dots, \frac{1}{h_n} \quad \text{is an AP}$

General Term ($n$-th Term)

Let the reciprocal AP have first term $A_1 = \frac{1}{h_1}$ and common difference $d = \frac{1}{h_2} - \frac{1}{h_1}$. The $n$-th term of the reciprocal AP is:

$A_n = \frac{1}{h_1} + (n-1)d$

Inverting $A_n$ gives the general term of the HP:

$h_n = \frac{1}{A_n} = \frac{1}{\frac{1}{h_1} + (n-1)d} = \frac{h_1 h_2}{h_2 + (n-1)(h_1 - h_2)}$

Important Examination Note: There is no closed-form algebraic formula for the sum of $n$ terms of a general harmonic progression ($S_n = \sum \frac{1}{a + kd}$). Never invert the sum formula of an AP to calculate the sum of an HP. Questions on ECAT invariably test term extraction, common differences in the reciprocal domain, or harmonic mean insertion.

Harmonic Mean (HM)

The Harmonic Mean ($H$) between two numbers $a$ and $b$ is the reciprocal of the arithmetic mean of their reciprocals:

$\frac{1}{H} = \frac{\frac{1}{a} + \frac{1}{b}}{2} = \frac{a+b}{2ab} \implies H = \frac{2ab}{a+b}$

To insert $n$ harmonic means between $a$ and $b$, insert $n$ arithmetic means between $\frac{1}{a}$ and $\frac{1}{b}$ with common difference $d = \frac{\frac{1}{b} - \frac{1}{a}}{n+1} = \frac{a-b}{(n+1)ab}$, and then invert each resulting term.

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5. Relationships Between Means ($A, G, H$)

For two positive, distinct real numbers $a$ and $b$ ($a \ne b$, $a, b > 0$), their three means are:

$A = \frac{a+b}{2}, \quad G = \sqrt{ab}, \quad H = \frac{2ab}{a+b}$

1. The Fundamental Identity ($G^2 = AH$)

Multiplying $A$ and $H$:

$A \cdot H = \left(\frac{a+b}{2}\right) \left(\frac{2ab}{a+b}\right) = ab = (\sqrt{ab})^2 = G^2$

Because $G^2 = AH$, the geometric mean $G$ is itself the geometric mean of the arithmetic mean $A$ and harmonic mean $H$. Thus, $A, G, H$ form a Geometric Progression in that order.

2. The Inequality of Means ($A \ge G \ge H$)

Evaluating the difference $A - G$:

$A - G = \frac{a+b}{2} - \sqrt{ab} = \frac{a + b - 2\sqrt{ab}}{2} = \frac{(\sqrt{a} - \sqrt{b})^2}{2}$

Because $(\sqrt{a} - \sqrt{b})^2 > 0$ for all real $a \ne b$, it follows that $A > G$.

Next, evaluating the difference $G - H$:

$G - H = \sqrt{ab} - \frac{2ab}{a+b} = \frac{\sqrt{ab}(a+b) - 2ab}{a+b} = \frac{\sqrt{ab}(a + b - 2\sqrt{ab})}{a+b} = \frac{\sqrt{ab}(\sqrt{a} - \sqrt{b})^2}{a+b}$

Since $a, b > 0$ and $a \ne b$, both numerator and denominator are strictly positive, so $G > H$.

Combining these inequalities:

$A > G > H \quad (a \ne b, \quad a, b > 0)$

If and only if $a = b$, the equality condition holds: $A = G = H = a$.

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6. Arithmetico-Geometric Series (AGS)

An Arithmetico-Geometric Series is formed by multiplying corresponding terms of an AP and a GP:

$S_n = a + (a+d)r + (a+2d)r^2 + (a+3d)r^3 + \dots + [a+(n-1)d]r^{n-1}$

Sum to Infinity ($S_\infty$ for $|r| < 1$)

To find the infinite sum, write the series and multiply by $r$:

$$\begin{aligned} S_\infty &= a + (a+d)r + (a+2d)r^2 + (a+3d)r^3 + \dots \\ r S_\infty &= \quad a r + (a+d)r^2 + (a+2d)r^3 + \dots \end{aligned}$$

Subtracting the second equation from the first:

$(1-r)S_\infty = a + d\left(r + r^2 + r^3 + \dots\right) = a + \frac{dr}{1-r}$

Dividing both sides by $(1-r)$ gives the standard closed-form formula:

$S_\infty = \frac{a}{1-r} + \frac{dr}{(1-r)^2} \quad (|r| < 1)$

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7. Sigma ($\Sigma$) Summation Identities for Natural Numbers

Many entrance examination problems require evaluating sums of polynomials $\sum_{k=1}^n P(k)$. These problems rely on the following closed-form identities:

Sum of the First $n$ Natural Numbers:

$\sum_{k=1}^{n} k = 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}$

Sum of the Squares of the First $n$ Natural Numbers:

$\sum_{k=1}^{n} k^2 = 1^2 + 2^2 + 3^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$

Sum of the Cubes of the First $n$ Natural Numbers:

$\sum_{k=1}^{n} k^3 = 1^3 + 2^3 + 3^3 + \dots + n^3 = \left[\frac{n(n+1)}{2}\right]^2 = \left(\sum_{k=1}^n k\right)^2$

Useful Linear Decompositions:

• Sum of first $n$ odd positive integers: $\sum_{k=1}^n (2k - 1) = 2\left[\frac{n(n+1)}{2}\right] - n = n^2$
• Sum of first $n$ even positive integers: $\sum_{k=1}^n 2k = 2\left[\frac{n(n+1)}{2}\right] = n(n+1)$

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8. Summary Comparison of Progressions