Class 9 › Mathematics › Chapter
Predicting What Comes Next: Sequences and Progressions
Sequences, explicit and recursive rules, arithmetic and geometric progressions, sum of natural numbers, fractal patterns and how AP and GP look on a graph.
1. What is a Sequence?
Sequence: an ordered list of numbers. Each number is a term of the sequence.
A sequence is a list where the position of each number matters. Knowing the pattern lets us predict the next terms.
- Finite sequence: has a last term. Infinite sequence: continues forever; three dots "$\dots$" show that.
- Notation: $t_1$ is the first term, $t_2$ the second, $t_n$ the $n$th term. The small number (subscript) is the position. So $t_4 = 7$ means the 4th term is $7$.
- Different sequences can use different letters ($t_n$, $s_n$, $u_n$, $V_n$).
- The position $n$ is a natural number ($1, 2, 3, \dots$), but the term itself can be negative, a fraction or any real number.
- Terms may increase, decrease, or alternate. Some sequences (like the primes) have no simple rule.
2. Some Important Number Sequences
| Sequence | Pattern | $n$th term |
|---|---|---|
| Natural numbers $1, 2, 3, \dots$ | Each term is $1$ more than the previous | $n$ |
| Odd numbers $1, 3, 5, \dots$ | Differences are always $2$ | $2n - 1$ |
| Square numbers $1, 4, 9, 16, \dots$ | Differences are $3, 5, 7, 9, \dots$ (odd numbers) | $n^2$ |
| Triangular numbers $1, 3, 6, 10, 15, \dots$ | Differences are $2, 3, 4, 5, \dots$ | $\dfrac{n(n+1)}{2}$ |
| Prime numbers $2, 3, 5, 7, 11, \dots$ | No simple rule to predict the next prime | None |
Each triangular number is a sum of natural numbers ($6 = 1+2+3$), drawn as a triangle of dots. Each square number is a sum of odd numbers ($n^2 = 1 + 3 + 5 + \dots + (2n-1)$), because adding the next odd number to a square of dots adds an L-shaped border.
- "Adding up all the terms so far" produces a new sequence (the sequence of running totals).
3. Explicit Rule for a Sequence
Explicit formula: a rule that gives the $n$th term directly from its position $n$, for example $u_n = 2n - 1$.
Put the position number into the formula and out comes the term. You do not need any earlier terms.
- Finding any term: substitute the position (the $53$rd term needs only $n = 53$).
- Checking whether a number is a term: put the formula equal to that number and solve for $n$.
- Finding its position: if the solution $n$ is a natural number, the number is a term and $n$ is its position. If $n$ is a fraction, zero or negative, it is not a term (positions are whole numbers starting from $1$).
- Finding the rule: look at how the terms change and guess a formula in $n$; verify it by substituting several values of $n$.
4. Recursive Rule for a Sequence
Recursive formula: a rule that gives each term using the previous term(s), together with the starting term(s). For example: $t_1 = 1$, $t_n = t_{n-1} + 3$ for $n \ge 2$.
A recursive rule is like a set of instructions: "start here, then repeat this step." You must know the earlier terms to get the next one.
| Explicit rule | Recursive rule | |
|---|---|---|
| Uses | Position $n$ | Previous term(s) |
| To find the 100th term | Substitute $n = 100$ directly | Must find all earlier terms first |
| Needs a starting value? | No | Yes (first term, or first few terms) |
- Computing terms: substitute $n = 2, 3, 4, \dots$ one by one. A recursive rule can be non-linear (for example $t_n = t_{n-1}(t_{n-1} - 1)$).
- More than one previous term: the Virahānka–Fibonacci sequence is $V_1 = 1$, $V_2 = 2$, $V_n = V_{n-1} + V_{n-2}$ giving $1, 2, 3, 5, 8, 13, 21, 34, \dots$ (each term is the sum of the previous two). Such a rule needs two starting terms.
- Rules can use three previous terms too (each term is the sum of the previous three).
- Virahānka (7th century) found it studying Prakrit poetry metre; Gopāla, Hemachandra and Fibonacci studied it later.
5. Arithmetic Progression (AP)
Arithmetic progression: a sequence in which the difference between consecutive terms is constant. This constant is the common difference $d$.
In an AP you keep adding the same number. Starting from $a$ and adding $d$ each time gives $a, a+d, a+2d, \dots$.
General form and $n$th term of an AP
$$a,\; a+d,\; a+2d,\; \dots,\; a+(n-1)d \qquad\Longrightarrow\qquad t_n = a + (n-1)d$$
- $a$ = first term, $d$ = common difference $= t_{k+1} - t_k$ (any term minus the one before it).
- $d \gt 0$: terms increase. $d \lt 0$: terms decrease. $d = 0$: all terms equal.
- Recursive rule: $t_1 = a$, $t_n = t_{n-1} + d$ for $n \ge 2$.
- Simplified: $t_n = dn + (a - d)$, which is a linear polynomial in $n$ (so APs are the linear patterns seen earlier).
Where $t_n = a + (n-1)d$ comes from
$t_1 = a$, $t_2 = a + 1\cdot d$, $t_3 = a + 2d$, $t_4 = a + 3d$. The $n$th term has the difference $d$ added $(n-1)$ times: one fewer than the position, because $t_1$ has no $d$ added.
Plotting $(n, t_n)$: an AP gives points on a straight line; a GP (ratio greater than $1$) rises more and more steeply.
- Graph of an AP: the points $(n, t_n)$ lie on a straight line whose slope is $d$ (this is why APs are the linear patterns of the earlier chapter).
- Testing for an AP: find $t_2 - t_1$, $t_3 - t_2$, ...; if all are equal, it is an AP. The natural numbers, odd numbers and the sequence of squares added by corner tiles are APs; the squares, triangular numbers and primes are not.
- Finding $a$ and $d$ from two terms: write each given term using $t_n = a + (n-1)d$ to get two linear equations in $a$ and $d$, and solve them (subtract to get $d$).
- Which term is a given number? Put $a + (n-1)d$ equal to it and solve for $n$ (must be a natural number).
- Number of terms in a list from first term $a$ to last term $l$: $n = \dfrac{l - a}{d} + 1$. This counts terms such as the multiples of a number in a range (first and last multiple give $a$ and $l$, and $d$ is the number itself).
- Real life: a fixed charge plus a constant rate per unit (taxi fare, fixed increments in salary) gives an AP; the fixed part acts like the constant term in $t_n = dn + (a - d)$.
6. Sum of the First $n$ Natural Numbers (and of an AP)
Sum of the first $n$ natural numbers
$$S_n = 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}$$
Reverse-and-add method
1. Write $S = 1 + 2 + \dots + n$ and again in reverse: $S = n + (n-1) + \dots + 1$.
2. Add the two lines column by column. Every pair adds to $n + 1$, and there are $n$ pairs: $2S = n(n+1)$.
3. So $S = \dfrac{n(n+1)}{2}$.
Two copies of the same staircase of dots fit together to make a rectangle that is $n$ dots tall and $(n+1)$ wide. The sum is half of that rectangle.
- Āryabhaṭa's form: sum $=$ (first term $+$ last term) $\div\, 2 \times$ number of terms, that is, average $\times$ number of terms.
- Sum of consecutive numbers from $m$ to $n$: $S_n - S_{m-1}$ (subtract the part before $m$).
- Triangular numbers: the $n$th triangular number is $S_n = \dfrac{n(n+1)}{2}$.
- Smallest $n$ so that the sum exceeds a value: solve $\dfrac{n(n+1)}{2} \gt$ value (try values of $n$ near the answer).
Sum of the first $n$ terms of an AP (same idea)
$$S_n = \frac{n}{2}\,(a + l) = \frac{n}{2}\,\bigl[\,2a + (n-1)d\,\bigr]$$
- $l$ = last term $= a + (n-1)d$. Reverse-and-add again: each pair of terms adds to $a + l$, and there are $n$ pairs, so $2S_n = n(a+l)$.
- Use it for "sum of all multiples of a number in a range": find the number of terms with $n = \frac{l-a}{d}+1$, then apply the formula.
7. Geometric Progression (GP)
Geometric progression: a sequence in which each term is obtained by multiplying the previous term by a fixed number $r$, the common ratio. Equivalently, the ratio of consecutive terms is constant.
In a GP you keep multiplying by the same number (an AP keeps adding). Doubling, tripling and "a fraction of the previous amount" are all GPs.
General form and $n$th term of a GP
$$a,\; ar,\; ar^2,\; ar^3,\; \dots,\; ar^{n-1} \qquad\Longrightarrow\qquad t_n = a\,r^{\,n-1}$$
- $a$ = first term, $r = \dfrac{t_{k+1}}{t_k}$ = common ratio.
- Recursive rule: $t_1 = a$, $t_n = r\,t_{n-1}$ for $n \ge 2$.
- The power of $r$ is one less than the position ($t_1$ has $r^0$).
| Common ratio | Behaviour of the terms |
|---|---|
| $r \gt 1$ | Terms grow faster and faster |
| $0 \lt r \lt 1$ | Terms shrink towards $0$ (never become negative) |
| $r = 1$ | All terms equal |
| $r = -1$ | Terms alternate between $a$ and $-a$ |
| $r \lt 0$ | Signs alternate (positive, negative, positive, ...) |
- Testing for a GP: compute $\dfrac{t_2}{t_1}, \dfrac{t_3}{t_2}, \dots$; if all are the same, it is a GP. $r$ may be a fraction, an integer or negative.
- Finding which term is a given number: put $a r^{n-1}$ equal to it and solve for $n$ by writing both sides as powers of the same base.
- Finding $a$ and $r$ from two terms: divide the two equations (so $a$ cancels) to get a power of $r$, then find $r$ and then $a$. If the $q$th term and $p$th term are known, $\dfrac{t_q}{t_p} = r^{\,q-p}$.
- Graph of a GP: the points $(n, t_n)$ do not lie on a straight line; they follow a curve that rises quickly for $r \gt 1$ and flattens towards $0$ for $0 \lt r \lt 1$.
- Real life: a quantity that doubles every hour (bacteria: $a \cdot 2^n$ after $n$ hours), or a bouncing ball that rises to a fixed fraction of its previous height (height after $n$ bounces $= H \cdot r^n$) both give GPs.
8. Fractals and GPs
Fractal: a shape or pattern that repeats itself at different scales: zooming into a small part looks like the whole. Fractals are made by repeating a simple rule (examples in nature: trees, cauliflower, snowflakes, coastlines).
At every stage each piece is replaced by the same smaller pattern. So the number of pieces is multiplied by a fixed number and the area is also multiplied by a fixed number, which is exactly what a GP does.
Sierpiński triangle
Rule
Join the midpoints of the sides of each black triangle and remove the centre one. Each black triangle becomes $3$ smaller ones, each $\tfrac14$ of its area.
Number of black triangles
$t_n = 3^n$
GP with ratio $3$: $1, 3, 9, 27, 81, \dots$ (stage $0$ has $1$). Recursive: multiply the previous count by $3$. It grows very fast.
Black area (stage 0 area $= 1$)
$s_n = \left(\tfrac34\right)^n$
GP with ratio $\tfrac34$: each stage keeps $\tfrac34$ of the previous area. The area decreases and gets closer and closer to $0$ as $n$ increases.
Sierpiński carpet
Square version
Divide each square into $9$ equal squares and remove the centre one, keeping $8$. Number of squares $= 8^n$; remaining area $= \left(\tfrac89\right)^n$ (also tends to $0$).
- General idea: if each piece is replaced by $k$ smaller pieces, each $\dfrac{1}{m}$ of the area, then the count is a GP with ratio $k$ and the area is a GP with ratio $\dfrac{k}{m}$.
- Note that the stage number can start from $0$ (stage $0$ is the original shape), so the $n$th stage and the $n$th term need care.
9. AP vs GP
| Arithmetic progression | Geometric progression | |
|---|---|---|
| Rule | Add a fixed number $d$ | Multiply by a fixed number $r$ |
| Constant quantity | Difference $t_{n+1} - t_n = d$ | Ratio $\dfrac{t_{n+1}}{t_n} = r$ |
| General form | $a, a+d, a+2d, \dots$ | $a, ar, ar^2, \dots$ |
| $n$th term | $a + (n-1)d$ | $a\,r^{n-1}$ |
| Recursive rule | $t_n = t_{n-1} + d$ | $t_n = r\,t_{n-1}$ |
| Graph of $(n, t_n)$ | Points on a straight line (slope $d$) | Points on a curve |
| Type of change | Same amount each step (linear growth or decay) | Same factor each step (very fast growth or shrinking) |
10. Formula Sheet and Common Mistakes
| Concept | Formula |
|---|---|
| AP: $n$th term | $t_n = a + (n-1)d$ |
| AP: number of terms | $n = \dfrac{l-a}{d} + 1$ |
| Sum of first $n$ natural numbers | $\dfrac{n(n+1)}{2}$ |
| Sum from $m$ to $n$ | $S_n - S_{m-1}$ |
| Sum of an AP | $\dfrac{n}{2}(a + l) = \dfrac{n}{2}[2a + (n-1)d]$ |
| GP: $n$th term | $t_n = a\,r^{n-1}$ |
| Triangular / square / odd numbers | $\dfrac{n(n+1)}{2}$ / $n^2$ / $2n-1$ |
| Fibonacci type | $V_n = V_{n-1} + V_{n-2}$ (two starting terms) |
Common mistakes:
• Using $n$ instead of $(n-1)$ in $a + (n-1)d$ and $ar^{n-1}$.
• Calling a sequence an AP after checking only one difference (check all of them).
• Forgetting that $d$ and $r$ can be negative (or $r$ a fraction).
• Accepting a non-natural-number $n$ as a position.
• Mixing up explicit and recursive formulas, or forgetting the starting term in a recursive rule.
• Confusing an AP (adds) with a GP (multiplies), or expecting a GP graph to be a straight line.
• In $S_n - S_{m-1}$, subtracting $S_m$ instead of $S_{m-1}$.
• Using $n$ instead of $(n-1)$ in $a + (n-1)d$ and $ar^{n-1}$.
• Calling a sequence an AP after checking only one difference (check all of them).
• Forgetting that $d$ and $r$ can be negative (or $r$ a fraction).
• Accepting a non-natural-number $n$ as a position.
• Mixing up explicit and recursive formulas, or forgetting the starting term in a recursive rule.
• Confusing an AP (adds) with a GP (multiplies), or expecting a GP graph to be a straight line.
• In $S_n - S_{m-1}$, subtracting $S_m$ instead of $S_{m-1}$.