Sequences and Series: Arithmetic and Geometric Progressions

Sequences and Series

1.1 Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference (d).

• General Term (nth term): $T_n = a + (n-1)d$

• Common Difference: $d = T_{n} - T_{n-1}$

• First Term: $a$

Example: For the sequence 2, 4, 6, 8, ...

• $a = 2$

• $d = 2$

• $T_n = 2 + (n-1)2$

Symbol Meanings

Symbol	Meaning
$a$	First term
$d$	Common difference
$n$	Number of terms
$T_n$	nth term
$S_n$	Sum of first n terms

1.2 The Sum of an Arithmetic Series

The sum of the first n terms of an arithmetic sequence (arithmetic series) is given by:

• $S_n = \frac{n}{2}(a + l)$, where $l$ is the last term

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

Proof Outline: Add the sequence forwards and backwards, then divide by 2.

Example: Find the sum of the first 20 terms of 3, 7, 11, ...

• $a = 3$, $d = 4$, $n = 20$

• $S_{20} = \frac{20}{2}[2 \times 3 + (20-1)4] = 10[6 + 76] = 10 \times 82 = 820$

1.3 Geometric Sequences

A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant called the common ratio (r).

• General Term (nth term): $T_n = a r^{n-1}$

• Common Ratio: $r = \frac{T_{n}}{T_{n-1}}$

Example: For the sequence 2, 6, 18, 54, ...

• $a = 2$

• $r = 3$

• $T_n = 2 \times 3^{n-1}$

2.2 The Sum of a Geometric Series

The sum of the first n terms of a geometric sequence is:

• $S_n = a \frac{1 - r^n}{1 - r}$, for $r \neq 1$

If $|r| < 1$, the series converges as $n \to \infty$:

• $S_{\infty} = \frac{a}{1 - r}$

Example: Find the sum of the first 5 terms of 3, 6, 12, ...

• $a = 3$, $r = 2$, $n = 5$

• $S_5 = 3 \frac{1 - 2^5}{1 - 2} = 3 \frac{1 - 32}{-1} = 3 \times 31 = 93$

3. Sigma Notation

Sigma notation ($\sum$) is used to represent the sum of a sequence. For example, $\sum_{k=1}^{n} (2k-1)$ represents the sum of the first n odd numbers.

• To evaluate, substitute values of $k$ from the lower to upper limit and sum the results.

Example: $\sum_{k=1}^{5} (2k-1) = 1 + 3 + 5 + 7 + 9 = 25$

4. Convergence of Series

A geometric series converges if $|r| < 1$. The sum to infinity is:

• $S_{\infty} = \frac{a}{1 - r}$

Example: For $a = 4$, $r = 0.5$, $S_{\infty} = \frac{4}{1 - 0.5} = 8$

5. Applications and Problem Types

• Finding the nth term or sum of arithmetic or geometric sequences

• Determining the number of terms needed to reach a sum

• Solving for unknowns in sequence formulas

• Using sigma notation to express and evaluate sums

• Identifying convergence and calculating sums to infinity for geometric series

6. Common Formulas

• Arithmetic nth term: $T_n = a + (n-1)d$

• Arithmetic sum: $S_n = \frac{n}{2}(a + l)$ or $S_n = \frac{n}{2}[2a + (n-1)d]$

• Geometric nth term: $T_n = a r^{n-1}$

• Geometric sum: $S_n = a \frac{1 - r^n}{1 - r}$

• Sum to infinity (geometric, $|r| < 1$): $S_{\infty} = \frac{a}{1 - r}$

7. Example Table: Arithmetic vs Geometric Sequences

Property	Arithmetic Sequence	Geometric Sequence
General Term	$T_n = a + (n-1)d$	$T_n = a r^{n-1}$
Sum of n Terms	$S_n = \frac{n}{2}(a + l)$	$S_n = a \frac{1 - r^n}{1 - r}$
Common Difference/Ratio	$d$	$r$
Convergence	Does not converge	Converges if $|r| < 1$

Additional info: These notes include both worked examples and exercises for practice, covering all standard types of problems involving arithmetic and geometric sequences and series, as well as sigma notation and convergence criteria. The material is foundational for probability and statistics, especially in understanding series, summation, and patterns in data.