Class 9 Maths Chapter 8: Predicting What Comes Next: Exploring Sequences and Progressions — Notes, Examples & Practice

A sequence is an ordered list of numbers $a_1, a_2, a_3, \dots$; the subscript is the position $n$ and $a_n$ is the term at that position. Position and value are different things — the 4th term of 5, 9, 13, 17 is 17.

An explicit rule gives any term straight from $n$, for example $a_n = 4n + 1$. You can jump to $a_{100}$ in one step. A recursive rule gives each term from the previous ones and needs starting value(s): $a_1 = 5$, $a_n = a_{n-1} + 4$. Both rules describe the same sequence here.

The Virahāṅka–Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, … has the recursive rule $a_n = a_{n-1} + a_{n-2}$. The Indian scholar Virahāṅka found it (around the 7th century) while counting rhythms made of short and long syllables, centuries before Fibonacci in Europe.

An arithmetic progression (AP) has a constant common difference $d = a_{n+1} - a_n$. Starting from the first term $a$, each step adds $d$, so after $n - 1$ steps: $a_n = a + (n - 1)d$.

$d > 0$ gives an increasing AP, $d < 0$ a decreasing one, $d = 0$ a constant sequence. Plotted as points $(n, a_n)$, an AP lies on a straight line — it is linear growth (Chapter 2) in disguise.

To test whether a list is an AP, compute every consecutive difference in the same order (later minus earlier) and check they are all equal.

To add $1 + 2 + \dots + n$, write the sum twice — forwards and backwards — and add the columns: every column gives $n + 1$, and there are $n$ columns. So twice the sum is $n(n + 1)$.

A picture shows the same thing: a staircase of $1, 2, \dots, n$ dots, together with a copy turned upside down, fills an $n \times (n + 1)$ rectangle. The numbers $1, 3, 6, 10, 15, \dots$ are called triangular numbers.

The same pairing idea works for any AP: the sum of an AP is (number of terms) × (average of first and last term).

A geometric progression (GP) has a constant non-zero common ratio $r = \dfrac{a_{n+1}}{a_n}$. Each step multiplies by $r$, so $a_n = a r^{n-1}$.

If $r > 1$ the terms grow faster and faster (doubling: 1, 2, 4, 8, …); if $0 < r < 1$ they shrink towards 0 (halving); a negative $r$ makes the signs alternate. Unlike an AP, the points $(n, a_n)$ of a GP do not lie on a straight line.

The Tower of Hanoi puzzle is a classic: moving $n$ discs takes $2^n - 1$ moves at least, because each extra disc more than doubles the work.

A fractal is a shape built by repeating the same step at smaller and smaller scales. Start with an equilateral triangle, join the midpoints of its sides and remove the middle triangle: you get 3 smaller triangles. Repeat on each: 9, then 27 … — the counts form a GP with ratio 3 (the Sierpiński triangle).

Each new triangle has $\dfrac{1}{4}$ of the area of its parent, so the shaded area shrinks by a factor of $\dfrac{3}{4}$ each stage: another GP.

Comparing an AP and a GP side by side — dots added in rows versus dots doubled — shows why multiplying eventually overtakes adding, however large the added amount.