Geometric Progression (GP): Mastering the Fundamentals

Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This concept is fundamental in advanced algebra and frequently appears in competitive exams like JEE.

Defining Geometric Progression

A sequence $a_1, a_2, a_3, \dots$ is a Geometric Progression if there exists a constant $r \neq 0$ such that $a_{n+1} = a_n \cdot r$ for all $n \geq 1$ . The first term is denoted by $a$ (or $a_1$ ), and the common ratio is denoted by $r$ .

What is the defining characteristic of a Geometric Progression?

Each term after the first is obtained by multiplying the previous term by a constant non-zero number called the common ratio.

The General Term of a GP

The nth term of a GP is $a \cdot r^{n-1}$.

Starting with the first term 'a', each subsequent term is found by multiplying by 'r'. So, the second term is $a \cdot r^1$ , the third is $a \cdot r^2$ , and generally, the nth term is $a \cdot r^{n-1}$ .

Let the first term of a GP be $a$ and the common ratio be $r$ . The terms of the GP are: $a_1 = a$ $a_2 = a \cdot r$ $a_3 = a_2 \cdot r = (a \cdot r) \cdot r = a \cdot r^2$ $a_4 = a_3 \cdot r = (a \cdot r^2) \cdot r = a \cdot r^3$ Following this pattern, the $n$ -th term, denoted by $a_n$ , is given by the formula: $a_n = a \cdot r^{n-1}$ .

Sum of the First n Terms of a GP

The sum of the first $n$ terms of a GP, denoted by $S_n$ , has two common formulas depending on the value of the common ratio $r$ .

Condition	Formula for $S_n$
$r \neq 1$	$S_n = \frac{a(r^n - 1)}{r - 1}$ or $S_n = \frac{a(1 - r^n)}{1 - r}$
$r = 1$	$S_n = na$

When $r=1$ , all terms in the GP are identical to the first term 'a'. Therefore, the sum of 'n' such terms is simply $n imes a$ .

Sum to Infinity of a GP

The sum to infinity exists only when $|r| < 1$, and is given by $S_{\infty} = \frac{a}{1-r}$.

If the common ratio 'r' is between -1 and 1 (exclusive), the terms of the GP get progressively smaller, approaching zero. This allows us to calculate an infinite sum.

For a GP to have a finite sum to infinity, the absolute value of the common ratio must be less than 1 (i.e., $|r| < 1$ ). In this case, as $n$ approaches infinity, $r^n$ approaches 0. The formula for the sum to infinity ( $S_{\infty}$ ) is derived from the sum of the first $n$ terms formula by taking the limit as $n o \infty$ : $S_{\infty} = \lim_{n \to \infty} \frac{a(1 - r^n)}{1 - r} = \frac{a(1 - 0)}{1 - r} = \frac{a}{1 - r}$ .

Under what condition does a Geometric Progression have a finite sum to infinity, and what is the formula?

A GP has a finite sum to infinity if $|r| < 1$ . The formula is $S_{\infty} = \frac{a}{1-r}$ .

Geometric Mean

The geometric mean (GM) of two positive numbers $a$ and $b$ is the number $x$ such that $a, x, b$ form a GP. This means $\frac{x}{a} = \frac{b}{x}$ , which leads to $x^2 = ab$ , so $x = \sqrt{ab}$ .

Visualizing the progression of terms in a GP. Consider a GP with first term $a=2$ and common ratio $r=3$ . The sequence is $2, 6, 18, 54, \dots$ . Each term is 3 times the previous one. This exponential growth is characteristic of GPs. The formula for the nth term, $a_n = a \cdot r^{n-1}$ , shows this exponential relationship clearly.

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Properties and Applications

GPs appear in various contexts, including compound interest calculations, population growth models, radioactive decay, and the analysis of algorithms. Understanding their properties is crucial for solving problems in these areas.

Name one real-world application where Geometric Progressions are used.

Compound interest, population growth, radioactive decay, etc.

Learning Resources

Geometric Progression - Wikipedia(wikipedia)

Provides a comprehensive overview of geometric progressions, including definitions, formulas, properties, and applications.

Geometric Sequences and Series - Khan Academy(video)

An introductory video lesson explaining the concept of geometric sequences and their common ratio.

Geometric Series - Brilliant.org(documentation)

Explains geometric series, including the sum to infinity, with clear examples and interactive elements.

Geometric Progression Formulas - Byju's(blog)

A detailed explanation of GP formulas, including the nth term and sum of n terms, with solved examples.

JEE Mathematics: Geometric Progression - Vedantu(blog)

Focuses on GP concepts relevant to competitive exams like JEE, with practice problems and strategies.

Understanding Geometric Progressions - Math is Fun(documentation)

A beginner-friendly explanation of geometric progressions with simple examples and interactive tools.

Geometric Progression Problems with Solutions - Toppr(blog)

Offers a collection of solved problems on geometric progressions, covering various difficulty levels.

The Sum of an Infinite Geometric Series - Mathematics LibreTexts(documentation)

A focused explanation on the convergence and calculation of the sum of infinite geometric series.

Geometric Mean - MathWorld Wolfram(documentation)

Provides a formal definition and properties of the geometric mean, including its relation to geometric progressions.

Geometric Progression Tutorial - Tutorialspoint(tutorial)

A step-by-step tutorial covering the basics of geometric progression, including formulas and examples.