Mathematics: Class 9 — Chapter 8: Predicting What Comes Next: Exploring Sequences and Progressions

A sequence is defined as an ordered list of numbers where each number is a term of the sequence. Sequences can be finite (having a limited number of terms) or infinite (continuing indefinitely, indicated by three dots $\dots$).

• Natural Numbers: $1, 2, 3, 4, 5, 6, \dots$
• Odd Numbers: $1, 3, 5, 7, 9, 11, \dots$
• Triangular Numbers: $1, 3, 6, 10, 15, 21, \dots$ where each term is the sum of natural numbers up to that term (e.g., $15 = 1 + 2 + 3 + 4 + 5$).
• Square Numbers: $1, 4, 9, 16, 25, 36, \dots$ where each term is the sum of consecutive odd numbers up to that term (e.g., $16 = 1 + 3 + 5 + 7$).

An explicit formula uses the term's position number, $n$, to calculate its value directly without needing to know the previous terms. We use notations like $t_n$, $s_n$, or $u_n$ to represent the $n^{\text{th}}$ term.

• Example: For the sequence of odd numbers, the explicit rule is $u_n = 2n - 1$. Substituting $n = 1, 2, 3$ gives $u_1 = 1$, $u_2 = 3$, $u_3 = 5$.
• Finding Position: To check if a number like $137$ is in the sequence, solve $2n - 1 = 137 \implies n = 69$. Since $69$ is a natural number, $137$ is the $69^{\text{th}}$ term.

A recursive rule or recursive formula defines a term by relating it to one or more previous terms. To use a recursive rule, the initial term(s) must be specified.

• Example: The sequence $1, 4, 7, 10, 13, \dots$ can be defined recursively as $t_1 = 1$, $t_n = t_{n-1} + 3$ for $n \ge 2$.
• Virahānka-Fibonacci Sequence: A famous recursive sequence where each term is the sum of the previous two terms: $V_1 = 1$, $V_2 = 2$, and $V_n = V_{n-1} + V_{n-2}$ for $n \ge 3$. This generates the sequence $1, 2, 3, 5, 8, 13, 21, 34, \dots$. It was first documented by the Indian scholar Virahānka in the 7th century CE.

An arithmetic progression (AP) is a sequence in which the difference between any two consecutive terms is a constant value. This constant difference is called the common difference ($d$).

• General Form of an AP: $a, a + d, a + 2d, a + 3d, \dots, a + (n - 1)d$, where $a$ is the first term and $d$ is the common difference.
• $n^{\text{th}}$ Term of an AP: $$t_n = a + (n - 1)d$$

The sum of the first $n$ natural numbers, denoted by $S_n$, is given by the formula:

This formula can be derived by writing the sum forward and backward, then adding the two equations together ($2S = n(n+1)$). Historically, this result was first recorded in India in Āryabhaၩa's Āryabhaၩĩya (Chapter 2, Verse 19).

A geometric progression (GP) is a sequence in which each term after the first is obtained by multiplying the previous term by a non-zero constant called the common ratio ($r$).

• General Form of a GP: $a, ar, ar^2, ar^3, \dots, ar^{n-1}$, where $a$ is the first term and $r$ is the common ratio.
• $n^{\text{th}}$ Term of a GP: $$t_n = a r^{n-1}$$