## Solution.

$$\begin{aligned} & \operatorname{tg} \alpha \cdot \operatorname{tg} \beta \leq \frac{1}{3} \Leftrightarrow \frac{\sin \alpha \sin \beta}{\cos \alpha \cos \beta} \leq \frac{1}{3} \Leftrightarrow \\ & \Leftrightarrow \frac{\frac{1}{2}(\cos (\alpha-\beta)-\cos (\alpha+\beta))}{\frac{1}{2}(\cos (\alpha-\beta)+\cos (\alpha+\beta))} \leq \frac{1}{3} \end{aligned}$$

$$\frac{(\cos (\alpha-\beta)-\cos (\alpha+\beta))}{(\cos (\alpha-\beta)+\cos (\alpha+\beta))} \leq \frac{1}{3}$$

Since $\alpha+\beta=60^{\circ}$, the inequality becomes

$$\frac{2 \cos (\alpha-\beta)-1}{2 \cos (\alpha-\beta)+1} \leq \frac{1}{3} \Leftrightarrow 4 \cos (\alpha-\beta) \leq 4, \cos (\alpha-\beta) \leq 1(\cos (\alpha-\beta) \geq 0)$$

The last inequality is true. This completes the proof.

## PROGRESSIONS

## BASIC CONCEPTS AND FORMULAS

## Arithmetic Progression

An arithmetic progression is a sequence in which the first term $a_{1}$ is given, and each subsequent term, starting from the second, is equal to the previous term plus a constant number $d$, called the common difference of the progression.

If the first term $a_{1}$ and the common difference $d$ of an arithmetic progression are given, the $n$-th term of the arithmetic progression is calculated by the formula

$$a_{n}=a_{1}+d(n-1) .$$

Formula (4.1) is called the general term formula of an arithmetic progression.

## Properties of the Terms of an Arithmetic Progression

1. Each middle term of an arithmetic progression is equal to the half-sum of the terms equally distant from it:

$$a_{k}=\frac{a_{k-1}+a_{k+1}}{2}, k=2,3 \ldots, n-1$$

2. In a finite arithmetic progression, the sums of terms equally distant from the ends are equal to each other and equal to the sum of the extreme terms:

$$a_{1}+a_{n}=a_{2}+a_{n-1}=a_{k}+a_{n-k+1}=\ldots=2 a_{1}+d(n-1)$$

## Sum of the First $n$ Terms of an Arithmetic Progression

The sum of the first $n$ terms of an arithmetic progression is equal to

$$S_{n}=\frac{a_{1}+a_{n}}{2} \cdot n$$

Considering (4.3), i.e., that $a_{1}+a_{n}=2 a_{1}+d(n-1)$, formula (4.4) can be written as

$$S_{n}=\frac{2 a_{1}+d(n-1)}{2} \cdot n$$

## Geometric Progression

A geometric progression is a sequence in which the first term $b_{1}$ is given, and each subsequent term, starting from the second, is equal to the previous term multiplied by a constant number $q$, called the common ratio of the progression.

If the first term $b_{1}$ and the common ratio $q$ of a geometric progression are given, the $n$-th term of the geometric progression is calculated by the formula

$$b_{n}=b_{1} q^{n-1}$$

Formula (4.6) is called the general term formula of a geometric progression.

## Properties of the Terms of a Geometric Progression

1. The square of each middle term of a geometric progression is equal to the product of the terms equally distant from it, i.e.,

$$b_{k}^{2}=b_{k-1} b_{k+1}, k=2,3 \ldots, n-1$$

2. In a finite geometric progression, the products of terms equally distant from the ends are equal to each other and equal to the product of the extreme terms:

$$b_{1} \cdot b_{n}=b_{2} \cdot b_{n-1}=b_{3} \cdot b_{n-2}=\ldots=b_{k} \cdot b_{n-k+1}=\ldots=b_{1}^{2} \cdot q^{n-1}$$

3. The product of the first $n$ terms of a geometric progression with positive terms is equal to the $n$-th root of the product of its extreme terms:

$$P_{n}=\sqrt{\left(b_{1} \cdot b_{n}\right)^{n}}$$

In the general case,

$$\left|P_{n}\right|=\sqrt{\left|b_{1} \cdot b_{n}\right|^{n}}$$

## Sum of the First $n$ Terms of a Geometric Progression

The sum of the first $n$ terms of a geometric progression is calculated by the formula

$$S_{n}=\frac{b_{1}-b_{n} q}{1-q}(q \neq 1)$$

Considering (4.6), i.e., that $b_{n}=b_{1} q^{n-1}$, formula (4.10) can be written as

$$S_{n}=\frac{b_{1}\left(1-q^{n}\right)}{1-q}$$

## Sum of the Terms of an Infinite Geometric Progression

The infinite numerical series formed by the terms of a geometric progression $b_{1}+b_{2}+b_{3}+\ldots+b_{n}+\ldots$ converges when $|q|<1$, and its sum $S$ is equal to

$$S=\frac{b_{1}}{1-q}$$

Formula (4.12) is also called the formula for the sum of the terms of an infinitely decreasing geometric progression.