Sequence and Series

Sequence and Series: Understanding Patterns

Mathematics often helps us understand and predict patterns. Sequences and series are fundamental tools for describing numerical patterns and their sums. By the end of this note, you should be able to confidently solve problems involving Arithmetic and Geometric Progressions.

Core Concepts

What is a Sequence? A sequence is an ordered list of numbers, often called terms, that follow a specific rule or pattern. Each number in the sequence is a term. We typically denote the first term as $t_1$ (or 'a'), the second as $t_2$, and the $n^{th}$ term as $t_n$. Example: The sequence 2, 4, 6, 8, ... shows a pattern where each term is 2 more than the previous one. The $n^{th}$ term can be found by the rule $t_n = 2n$.

What is a Series? A series is simply the sum of the terms in a sequence. If you have a sequence $t_1, t_2, t_3, ...$, the corresponding series is $t_1 + t_2 + t_3 + ...$. The sum of the first 'n' terms is usually denoted as $S_n$. Example: For the sequence 2, 4, 6, 8, ..., the series for the first four terms would be $S_4 = 2 + 4 + 6 + 8 = 20$.

Arithmetic Progression (AP)

An Arithmetic Progression (AP) is a special type of sequence where the difference between consecutive terms is constant. This constant difference is known as the common difference (d). You can always find 'd' by subtracting any term from its preceding term ($d = t_n - t_{n-1}$). Example: In the sequence 3, 7, 11, 15, ..., the common difference 'd' is $7 - 3 = 4$.

Key Formulas for AP: To solve problems involving AP, you need to identify the first term ('a'), the common difference ('d'), and the number of terms ('n').

• The $n^{th}$ term ($t_n$): This formula helps you find any term in the sequence without listing all previous terms. $t_n = a + (n-1)d$
• The Sum of the first 'n' terms ($S_n$): This formula calculates the total sum of a certain number of terms. $S_n = \frac{n}{2} [2a + (n-1)d]$ Alternatively, if the last term ($t_n$) is known: $S_n = \frac{n}{2} [a + t_n]$

Example (AP): Let's find the $10^{th}$ term and the sum of the first 10 terms of the AP: 5, 8, 11, ...

1. Identify: First term $a = 5$. Common difference $d = 8 - 5 = 3$. We want $n = 10$.
2. $10^{th}$ term: $t_{10} = a + (10-1)d = 5 + 9(3) = 5 + 27 = 32$.
3. Sum of first 10 terms: $S_{10} = \frac{10}{2} [2(5) + (10-1)3] = 5 [10 + 9(3)] = 5 [10 + 27] = 5 [37] = 185$.

Geometric Progression (GP)

A Geometric Progression (GP) is a sequence where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio (r). You can find 'r' by dividing any term by its preceding term ($r = \frac{t_n}{t_{n-1}}$). Example: In the sequence 2, 6, 18, 54, ..., the common ratio 'r' is $\frac{6}{2} = 3$.

Key Formulas for GP: Similar to AP, identify the first term ('a'), the common ratio ('r'), and the number of terms ('n').

• The $n^{th}$ term ($t_n$): This formula allows you to find any term in the sequence quickly. $t_n = ar^{n-1}$
• The Sum of the first 'n' terms ($S_n$): This formula calculates the total sum of a given number of terms.
  • $S_n = \frac{a(r^n - 1)}{r - 1}$, when the common ratio $r > 1$.
  • $S_n = \frac{a(1 - r^n)}{1 - r}$, when the common ratio $r < 1$ (and $r \neq 1$).

Example (GP): Let's find the $5^{th}$ term and the sum of the first 5 terms of the GP: 3, 6, 12, ...

1. Identify: First term $a = 3$. Common ratio $r = \frac{6}{3} = 2$. We want $n = 5$.
2. $5^{th}$ term: $t_5 = ar^{5-1} = 3(2^4) = 3(16) = 48$.
3. Sum of first 5 terms: Since $r = 2$ (which is $> 1$), we use the first sum formula. $S_5 = \frac{3(2^5 - 1)}{2 - 1} = \frac{3(32 - 1)}{1} = 3(31) = 93$.

Real-World Example

Financial Growth: Imagine you invest N10,000 that earns 5% compound interest annually. The value of your investment each year forms a Geometric Progression, demonstrating how your money grows over time.

Practice/Reflection Question

A theatre has 20 seats in the first row, 22 in the second, 24 in the third, and so on. If there are 15 rows in total, how many seats are in the $15^{th}$ row? What is the total seating capacity of the theatre?