Arithmetico-Geometric Series

Algebra Progress

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Introduction to the arithmetico-geometric series or (AGS) and its appearance in contests.

Author: Jaden Park

Introduction

An arithmetic-geometric sequence is a sequence where every term can be written as the product of the terms of an arithmetic sequence and a geometric sequence.

An AGS looks like

a, (a + d)r, (a + 2d)r^{2}, (a + 3d)r^{3}, \ldots, [a + (n - 1)d]r^{n - 1}

where $a$ is the initial term, $d$ is the common difference, and $r$ is the common ratio.

The General term

The $n$ th term of the AGS is

t_n = [a + (n - 1)d]r^{n - 1}.

Sum of an AGS

The sum of the first $n$ terms of an AGS is

S_n = \frac{a - [a + (n - 1)d]r^{n}}{1 - r} + \frac{dr(1 - r^{n - 1})}{(1 - r)^{2}}.

Sum of an infinite AGS

If $|r| < 1,$ then the sum of an infinite AGS is

S_{\infty} = \frac{a}{1 - r} + \frac{dr}{(1 - r)^{2}}.

Here are some practice problems.

Practice Problem

Find the value of $x$ given that

3 + \frac{1}{4}(3 + x) + \frac{1}{4^{2}}(3 + 2x) + \frac{1}{4^{3}}(3 + 3x) + \ldots = 8.

Practice Problem

Find the positive integer $n$ for which

\left\lfloor \log_{2}{1} \right\rfloor + \left\lfloor \log_{2}{2} \right\rfloor + \left\lfloor \log_{2}{3} \right\rfloor + \ldots + \left\lfloor \log_{2}{n} \right\rfloor = 1994

(For real $x, \left\lfloor x \right\rfloor$ is the greatest integer $\leq x.$ )

Final Notes and Tips

Not that common, personally wouldn't memorize the formula.

Harmonic Series

Telescoping Series

Algebra Progress

Polynomials

Sequence and Series

Miscellaneous

Introduction

The General term

Sum of an AGS

Sum of an infinite AGS

Practice Problem

Practice Problem

Final Notes and Tips

Introduction

The General term

Sum of an AGS

Sum of an infinite AGS

Practice Problem

Practice Problem

Final Notes and Tips