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

We see patterns around us everywhere in nature, art, music, and finance. In mathematics, a sequence is an ordered list of numbers where each number is called a term of the sequence.

• Natural Numbers: $1, 2, 3, 4, 5, 6, \dots$
  • Every term is one more than the previous term.
• Odd Numbers: $1, 3, 5, 7, 9, 11, \dots$
  • There is a constant difference of $2$ between any two consecutive terms.
• Triangular Numbers: $1, 3, 6, 10, 15, 21, \dots$
  • Each term is the sum of the natural numbers up to that term. For example, the fifth triangular number is $15 = 1 + 2 + 3 + 4 + 5$.
• Square Numbers: $1, 4, 9, 16, 25, 36, \dots$
  • Each term is the sum of consecutive odd numbers up to that term. For example, $16 = 1 + 3 + 5 + 7$.

Sequences can be finite (having a limited number of terms, e.g., $6, 12, 24, 48, 96$) or infinite (continuing indefinitely, indicated by three dots $\dots$).

Notation: We use subscripts to match term numbers. $t_1$ represents the first term, $t_2$ represents the second term, and $t_n$ represents the $n^{\text{th}}$ term. Different letters like $s_n$ or $u_n$ can be used to distinguish different sequences.

An explicit formula (or explicit rule) uses a term's position number, $n$, to calculate its value directly without needing to know the previous terms.

• Example 1: For the sequence of odd numbers, the explicit rule is $u_n = 2n - 1$. Substituting $n = 53$ gives the $53^{\text{rd}}$ term: $u_{53} = 2(53) - 1 = 105$.
• Example 2: For the sequence generated by $s_n = 5n - 2$, we can check if $308$ is a term by solving $5n - 2 = 308 \implies 5n = 310 \implies n = 62$. Since $62$ is a natural number, $308$ is the $62^{\text{nd}}$ term.

A recursive rule (or recursive formula) defines a term by relating it to one or more of its preceding 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: $t_1 = 1$, and $t_n = t_{n-1} + 3$ for $n \ge 2$.

An arithmetic progression (AP) is a sequence in which the difference between consecutive terms is constant. 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.
• Explicit Formula for the $n^{\text{th}}$ Term:

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

• Recursive Rule: $t_1 = a$, and $t_n = t_{n-1} + d$ for $n \ge 2$.

The sum of the first $n$ natural numbers, denoted by $S_n = 1 + 2 + 3 + \dots + n$, is given by the formula:

$$S_n = \frac{n(n + 1)}{2}$$

• Derivation: By writing the sum forward and backward, and adding the two equations:

  $$S = 1 + 2 + \dots + n$$

  $$S = n + (n-1) + \dots + 1$$

  $$2S = n(n+1) \implies S = \frac{n(n+1)}{2}$$

• Historical Note: The first known written mention of this result is found in Āryabhaṭa's Āryabhaṭīya (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 constant factor 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.
• Explicit Formula for the $n^{\text{th}}$ Term:

  $$t_n = a \cdot r^{n-1}$$

• Recursive Rule: $t_1 = a$, and $t_n = r \cdot t_{n-1}$ for $n \ge 2$.