Chapter 5: Progressions

5.1 Arithmetic Progressions (AP)

1. Definition of Arithmetic Progression

An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is a constant, called the common difference ($d$).

$$d = T_n - T_{n-1}$$

where $T_n$ represents the $n$-th term.

2. The $n$-th Term of an AP

The general formula for the $n$-th term ($T_n$) of an AP is:

$$T_n = a + (n - 1)d$$

where:

• $a = T_1$: First term of the sequence
• $d$: Common difference
• $n$: Number of terms ($n \in \mathbb{Z}^+$)

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

The sum of the first $n$ terms of an AP can be calculated using either of two formulas:

$$S_n = \frac{n}{2} [2a + (n - 1)d]$$ $$S_n = \frac{n}{2} [a + l]$$

where $l = T_n$ is the last term of the sequence.

4. Relationship Between $T_n$ and $S_n$

The $n$-th term can be expressed in terms of sum of terms as:

$$T_n = S_n - S_{n-1} \quad (n \ge 2)$$

Note that $T_1 = S_1$.

5.2 Geometric Progressions (GP)

1. Definition of Geometric Progression

A Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio ($r$).

$$r = \frac{T_n}{T_{n-1}}$$

2. The $n$-th Term of a GP

The general formula for the $n$-th term ($T_n$) of a GP is:

$$T_n = a r^{n-1}$$

where $a$ is the first term and $r$ is the common ratio ($r \neq 0$).

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

The sum of the first $n$ terms of a GP is given by:

$$S_n = \frac{a(1 - r^n)}{1 - r} \quad (|r| < 1 \text{ or } r < 1)$$ $$S_n = \frac{a(r^n - 1)}{r - 1} \quad (|r| > 1 \text{ or } r > 1)$$

where $r \neq 1$.

4. Sum to Infinity ($S_\infty$)

A geometric progression converges to a finite sum as $n \to \infty$ if and only if the common ratio satisfies:

$$-1 < r < 1 \quad (|r| < 1)$$

The sum to infinity ($S_\infty$) is calculated using:

$$S_\infty = \frac{a}{1 - r}$$

Logically, if $|r| \ge 1$, the series diverges and $S_\infty$ does not exist.

5. Applications of Progressions

• Solving real-world scenarios involving financial interest, salary increments, geometric population growth, depreciation, repeating decimals ($0.\dot{3}\dot{6} = 0.36 + 0.0036 + \dots$), and bouncing ball distances.