Sequences and Progressions Class 9: AP, GP, Rules and Solved Questions

What Sequences and Progressions is about

Weightage: This chapter is part of Unit II, Algebra, which carries 20 of the 80 marks in the school annual exam, shared with Introduction to Polynomials, Exploring Algebraic Identities and Linear Equations in Two Variables.Source: CBSE Mathematics syllabus, Class IX 2026-27, course structure, Unit II Algebra

Key concepts

Sequences and terms

NCERT section 8.1 Introduction to Sequences

A sequence is a list of numbers in a definite order. Each number is a term; the first is $t_1$, the second $t_2$, and the n-th is $t_n$. Three dots (…) mean the list goes on forever.

Familiar examples:

• natural numbers 1, 2, 3, 4, …
• odd numbers 1, 3, 5, 7, …
• triangular numbers 1, 3, 6, 10, 15, … (add 2, 3, 4, 5, …)
• square numbers 1, 4, 9, 16, 25, …

Not every sequence has an easy rule: nobody has a simple formula that produces the prime numbers.

Worked example

Write the next two terms of each sequence and describe the pattern: (i) 1, 3, 6, 10, 15, … (ii) 2, 6, 18, 54, … (iii) 40, 35, 30, 25, …

Try it first, then show the solutionSolution

1. (i) The gaps are 2, 3, 4, 5, so next add 6 and 7: 21, 28 (triangular numbers).
2. (ii) Each term is 3 times the previous: 162, 486.
3. (iii) Each term is 5 less: 20, 15.

Answer: (i) 21, 28 (ii) 162, 486 (iii) 20, 15

Explicit rule: the n-th term from n

NCERT section 8.2 Explicit Rule for a Sequence

An explicit rule (general rule) gives any term directly from its position n, for example $t_n=2n-1$ for odd numbers.

Uses:

• Find a term: put in the value of n.
• Which term is a value? Set $t_n$ equal to the value and solve for n.
• Is a number a term? Solve for n; the number is a term only if n comes out as a natural number.

Worked example

For $t_n=4n+3$: (i) find the 20th term; (ii) which term is 203? (iii) is 250 a term?

Try it first, then show the solutionSolution

1. (i) $t_{20}=4(20)+3=83$.
2. (ii) $4n+3=203$ gives $4n=200$, $n=50$: the 50th term.
3. (iii) $4n+3=250$ gives $n=61.75$, not a natural number, so 250 is not a term.

Answer: (i) 83 (ii) the 50th term (iii) no

Recursive rule and the Virahānka–Fibonacci sequence

NCERT section 8.3 Recursive Rule for a Sequence

A recursive rule gives the first term(s) and a rule for each term using earlier terms. For 1, 4, 7, 10, …: $t_1=1$, $t_n=t_{n-1}+3$ for $n\ge 2$.

To use a recursive rule you must work through the terms in order, since each one needs the one before.

A rule can use more than one earlier term. The Virahānka–Fibonacci sequence has $V_1=1$, $V_2=2$, $V_n=V_{n-1}+V_{n-2}$: 1, 2, 3, 5, 8, 13, 21, 34, … Virahānka (7th century CE) found it while counting rhythms in poetry; Gopāla and Hemachandra studied it later, and Fibonacci after them.

Worked example

(i) Find the first four terms of $u_1=2$, $u_n=3u_{n-1}-1$. (ii) Write the first ten terms of the Virahānka–Fibonacci sequence.

Try it first, then show the solutionSolution

1. (i) $u_2=3(2)-1=5$, $u_3=3(5)-1=14$, $u_4=3(14)-1=41$.
2. (ii) Start 1, 2 and keep adding the last two: 3, 5, 8, 13, 21, 34, 55, 89.

Answer: (i) 2, 5, 14, 41 (ii) 1, 2, 3, 5, 8, 13, 21, 34, 55, 89

Arithmetic progression and its n-th term

NCERT section 8.4 Arithmetic Progressions

An arithmetic progression (AP) adds a fixed number d, the common difference, to get each new term:
$a,a+d,a+2d,\dots t_n=a+(n-1)d$
To get to the n-th term you add d only $(n-1)$ times, because the first term already starts the list.

Recursive rule: $t_1=a$, $t_n=t_{n-1}+d$.

d can be positive (increasing AP), negative (decreasing) or zero (constant). Find it as $d=t_2-t_1$, and check with another pair.

Worked example

For the AP 7, 11, 15, 19, …, find the 30th term and write both rules.

Try it first, then show the solutionSolution

1. $a=7$, $d=11-7=4$.
2. $t_{30}=7+29\times 4=123$.
3. Explicit: $t_n=7+(n-1)4=4n+3$. Recursive: $t_1=7$, $t_n=t_{n-1}+4$.

Answer: $t_{30}=123$; $t_n=4n+3$; $t_1=7$, $t_n=t_{n-1}+4$.

Finding an AP from two of its terms

NCERT section 8.4 Arithmetic Progressions

If two terms are given, write each as $a+(n-1)d$ to get two linear equations in a and d. Subtracting removes a and gives d directly, because the gap between the m-th and n-th terms is $(n-m)d$.

This is how you solve 'the 3rd term is 12 and the 50th is 106' or salary problems where you know the pay in two different years.

Worked example

The 5th term of an AP is 23 and the 12th term is 51. Find the AP and its 30th term.

Try it first, then show the solutionSolution

1. $a+4d=23$ and $a+11d=51$.
2. Subtract: $7d=28$, so $d=4$; then $a=23-16=7$.
3. AP: 7, 11, 15, 19, …
4. $t_{30}=7+29\times 4=123$.

Answer: 7, 11, 15, …; 30th term 123

Visualising an AP as points on a line

NCERT section 8.4 Arithmetic Progressions

Plot each term as a point $(n,t_n)$. For an AP, $t_n=a+(n-1)d=dn+(a-d)$, which is a linear expression in n. So all the points lie on one straight line with slope d, just like the linear patterns of Chapter 2.

A GP does not do this: its points curve upwards (or flatten towards zero) because each step multiplies rather than adds.

Worked example

Plot the first four terms of $t_n=3n-1$ as points. What line do they lie on, and what is its slope?

Try it first, then show the solutionSolution

1. Terms: $t_1=2$, $t_2=5$, $t_3=8$, $t_4=11$.
2. Points: $(1,2)$, $(2,5)$, $(3,8)$, $(4,11)$.
3. They lie on $y=3x-1$; the slope 3 equals the common difference.

Answer: On $y=3x-1$, slope 3 = d.

Sum of the first n natural numbers

NCERT section 8.5 Sum of the First n Natural Numbers

Write the sum forwards and backwards and add:

• $S=1+2+\cdots +n$
• $S=n+(n-1)+\cdots +1$

Each of the n columns adds to $n+1$, so $2S=n(n+1)$ and
$S_n=\frac{n(n+1)}{2}$
In a picture, two staircases of dots fit together into an $n\times (n+1)$ rectangle. Āryabhaṭa's Āryabhaṭīya (Chapter 2, verse 19) gives this sum, including the idea 'average of first and last term × number of terms'.

The n-th triangular number is $\frac{n(n+1)}{2}$. For a run of consecutive numbers, subtract two sums: $25+\cdots +58=S_{58}-S_{24}$.

Worked example

Find (i) $1+2+\cdots +100$ (ii) $31+32+\cdots +70$.

Try it first, then show the solutionSolution

1. (i) $S_{100}=\frac{100\times 101}{2}=5050$.
2. (ii) $S_{70}-S_{30}=\frac{70\times 71}{2}-\frac{30\times 31}{2}=2485-465=2020$.

Answer: (i) 5050 (ii) 2020

Geometric progression and its n-th term

NCERT section 8.6 Geometric Progressions

A geometric progression (GP) multiplies by a fixed number r, the common ratio, to get each new term:
$a,ar,a{r}^2,\dots t_n=a{r}^{n-1}$
Recursive rule: $t_1=a$, $t_n=r\times t_{n-1}$. Find r as $\frac{t_2}{t_1}$.

If $r>1$ the terms grow very fast (bacteria doubling); if $0<r<1$ they shrink towards 0 (a bouncing ball, the area of a fractal). To find which term equals a value, divide by a and write the result as a power of r.

Worked example

(i) Find the 10th term of 3, 6, 12, … (ii) Which term of 5, 15, 45, … is 3645?

Try it first, then show the solutionSolution

1. (i) $a=3$, $r=2$: $t_{10}=3\times 2^9=1536$.
2. (ii) $a=5$, $r=3$: $5\times 3^{n-1}=3645$, so $3^{n-1}=729=3^6$.
3. $n-1=6$, so $n=7$.

Answer: (i) 1536 (ii) the 7th term

Fractals: the Sierpiński triangle and carpet

NCERT section 8.6 Geometric Progressions

A fractal is a pattern that looks similar to itself at every scale. The Sierpiński triangle: start with a filled triangle (Stage 0); join the midpoints of the sides and remove the middle triangle; repeat on every remaining triangle.

• Number of black triangles: 1, 3, 9, 27, … $=3^n$ (a GP with $r=3$).
• Black area (Stage 0 area 1): 1, $\frac{3}{4}$, ${\left(\frac{3}{4}\right)}^2$, … $={\left(\frac{3}{4}\right)}^n$ (a GP with $r=\frac{3}{4}$).

So the number of pieces grows without limit while the area shrinks towards 0.

The Sierpiński carpet cuts a square into 9 and removes the centre: $8^n$ squares and area ${\left(\frac{8}{9}\right)}^n$.

Worked example

For the Sierpiński triangle with Stage 0 area 1 square unit, find the number of black triangles and the black area at Stage 4. Write both rules.

Try it first, then show the solutionSolution

1. Triangles: $t_n=3^n$, so $t_4=81$.
2. Area: $s_n={\left(\frac{3}{4}\right)}^n$, so $s_4=\frac{81}{256}$ (about 0.32).
3. Recursive: $t_0=1$, $t_n=3t_{n-1}$; $s_0=1$, $s_n=\frac{3}{4}{s}_{n-1}$.

Answer: 81 triangles; area $\frac{81}{256}$ square units.

The Tower of Hanoi

The CBSE syllabus lists the Tower of Hanoi puzzle with this chapter. There are three pegs and n discs of different sizes stacked on one peg, largest at the bottom. Move the whole stack to another peg, one disc at a time, never placing a bigger disc on a smaller one.

To move n discs: move the top $n-1$ to the spare peg, move the biggest disc, then move the $n-1$ discs on top of it. So the least number of moves $M_n$ satisfies the recursive rule
$M_1=1,M_n=2M_{n-1}+1$
which gives 1, 3, 7, 15, 31, … The explicit rule is $M_n=2^n-1$: each term is one less than a term of the GP 2, 4, 8, 16, …

Worked example

Find the least number of moves for 4 and for 6 discs, using the recursive rule and then the explicit rule.

Try it first, then show the solutionSolution

1. Recursive: $M_1=1$, $M_2=3$, $M_3=7$, $M_4=15$, $M_5=31$, $M_6=63$.
2. Explicit: $M_4=2^4-1=15$ and $M_6=2^6-1=63$.

Answer: 15 moves for 4 discs and 63 moves for 6 discs.

Maths in real life: Sequences and Progressions

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Topics in this chapter

• Arithmetic Progressions: nth Term and Sum of Terms
• Geometric Progressions, Recursive Rules and Fractal Patterns

Sequences and Progressions formula sheet

n-th term of an AP

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

a first term, d common difference.

Recursive rule of an AP

$t_1=a,t_n=t_{n-1}+d$

Common difference

$d=t_2-t_1=t_{n+1}-t_n$

Gap between two terms

$t_n-t_m=(n-m)d$

Sum of the first n natural numbers

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

Also the n-th triangular number.

Sum of a run of numbers

$m+(m+1)+\cdots +n=S_n-S_{m-1}$

n-th term of a GP

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

a first term, r common ratio.

Recursive rule of a GP

$t_1=a,t_n=r\cdot t_{n-1}$

Common ratio

$r=\frac{t_2}{t_1}=\frac{t_{n+1}}{t_n}$

Virahānka–Fibonacci

$V_1=1,V_2=2,V_n=V_{n-1}+V_{n-2}$

Sierpiński triangle

$\text{triangles}=3^n,\text{area}={\left(\frac{3}{4}\right)}^n$

Sierpiński carpet

$\text{squares}=8^n,\text{area}={\left(\frac{8}{9}\right)}^n$

Tower of Hanoi

$M_1=1,M_n=2M_{n-1}+1;M_n=2^n-1$

How the school exam asks Sequences and Progressions

The question patterns to expect, and the method that works for each.

Tricks to remember Sequences and Progressions

Short memory aids. Say each one aloud once, then use it on a question the same day.

• AP adds, GP multiplies

  Arithmetic = Add; Geometric = Grow by multiplying.

  Use it when: Deciding which formula to use.

• n − 1 jumps

  To reach the n-th stair from the first you take only n − 1 steps.

  Use it when: Remembering $a+(n-1)d$ and $a{r}^{n-1}$.

• Forwards plus backwards

  Write 1 to n forwards and backwards; every pair makes n + 1.

  Use it when: Recalling and proving $\frac{n(n+1)}{2}$.

• Double and one more

  Tower of Hanoi: double the last answer and add one (1, 3, 7, 15, …).

  Use it when: The recursive rule $M_n=2M_{n-1}+1$.

• Natural or not?

  If solving for n gives a fraction or a negative number, the value is not a term.

  Use it when: 'Is this number a term?' questions.

Sequences and Progressions: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section E style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 4-mark question split into 1 + 1 + 2 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Sequences and Progressions

The slips examiners see most often, and what to write instead.

• ✗Mistake: Using a + nd instead of a + (n − 1)d

  Students count jumps from 0 instead of from the first term.

  Wrong working

  30th term of 7, 11, 15, …: $7+30\times 4=127$

  Right working

  $7+29\times 4=123$

  ✓Do this: Check your formula on $n=1$: it must give a.

  Usually costs: The answer mark.

• ✗Mistake: Wrong sign of d for a decreasing AP

  d is found as first minus second.

  Wrong working

  21, 18, 15, …: $d=21-18=3$

  Right working

  $d=18-21=-3$

  ✓Do this: Always compute later term minus earlier term.

  Usually costs: Every term found from d.

• ✗Mistake: Accepting a fractional n

  Students stop after solving the equation.

  Wrong working

  $6n-1=301$ gives $n=50.33$, so 301 is the 50th term.

  Right working

  n must be a natural number; 50.33 is not, so 301 is not a term.

  ✓Do this: State the conclusion from the type of n.

  Usually costs: 1 to 2 marks.

• ✗Mistake: Using r^n instead of r^(n − 1)

  The same counting slip as in APs.

  Wrong working

  10th term of 3, 6, 12, …: $3\times 2^{10}=3072$

  Right working

  $3\times 2^9=1536$

  ✓Do this: Test the formula on the first term: $a{r}^0=a$.

  Usually costs: The answer mark.

• ✗Mistake: Subtracting the wrong sum for a run

  The lower limit is included by mistake.

  Wrong working

  $25+\cdots +58=S_{58}-S_{25}$

  Right working

  $S_{58}-S_{24}=1711-300=1411$ (25 must stay in the sum)

  ✓Do this: Subtract the sum up to one less than the first number.

  Usually costs: The final answer.

NCERT exercise guide: Sequences and Progressions

What each exercise in the NCERT chapter tests, how to approach it, and a question of the same kind solved step by step. Questions here are written in our own words.

Practice questions on Sequences and Progressions

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   Which of these is a geometric progression?
   (a) 2, 4, 6, 8
   (b) 2, 6, 18, 54
   (c) 1, 4, 9, 16
   (d) 1, 2, 3, 5

   Hint

   Look for a constant ratio.

   Answer

   (b), ratio 3

2. Q2 · 1 mark · MCQExam level

   The 20th term of the AP 5, 9, 13, … is
   (a) 80
   (b) 81
   (c) 85
   (d) 76

   Hint

   $a+19d$.

   Answer

   (b) 81

3. Q3 · 1 mark · Assertion-reasonExam level

   Assertion (A): The 10th triangular number is 55.
   Reason (R): The n-th triangular number is $\frac{n(n+1)}{2}$.
   (a) Both A and R are true and R is the correct explanation of A.
   (b) Both A and R are true but R is not the correct explanation of A.
   (c) A is true but R is false.
   (d) A is false but R is true.

   Hint

   $\frac{10\times 11}{2}$.

   Answer

   (a)

4. Q4 · 2 marks · Very short answerExam level

   Write the first five terms of $t_n=2n^2-1$. Is it an AP?

   Hint

   Check the differences.

   Answer

   1, 7, 17, 31, 49. Differences 6, 10, 14, 18 are not constant, so it is not an AP.

5. Q5 · 2 marks · Very short answerHarder exam level

   Is 301 a term of the AP 5, 11, 17, …? Justify.

   Hint

   $t_n=6n-1$.

   Answer

   $6n-1=301$ gives $n=50\frac{1}{3}$, not a natural number, so 301 is not a term.

6. Q6 · 3 marks · Short answerHarder exam level

   A sequence has $t_1=-4$ and $t_{n+1}=t_n+5$. Find its first five terms and decide which term is 96.

   Hint

   It is an AP.

   Answer

   −4, 1, 6, 11, 16; $-4+5(n-1)=96$ gives $n=21$.

7. Q7 · 3 marks · Short answerHarder exam level

   The 4th term of an AP is 19 and its 9th term is 12 more than its 6th term. Find the AP and its 20th term.

   Hint

   $t_9-t_6=3d$.

   Answer

   $d=4$, $a=7$: 7, 11, 15, …; $t_{20}=7+76=83$.

8. Q8 · 2 marks · Very short answerHarder exam level

   Find $25+26+\cdots +60$.

   Hint

   $S_{60}-S_{24}$.

   Answer

   $1830-300=1530$

9. Q9 · 2 marks · Very short answerHarder exam level

   A culture has 50 bacteria and the number doubles every hour. How many are there after 6 hours and after n hours?

   Hint

   GP with $r=2$.

   Answer

   $50\times 2^6=3200$; $50\times 2^n$ after n hours.

10. Q10 · 5 marks · Long answerHarder exam level

    For the Sierpiński triangle (Stage 0 area 1), find the number of black triangles and the black area at Stage 5, and write the explicit and recursive rules for each.

    Hint

    Ratios 3 and $\frac{3}{4}$.

    Answer

    Triangles $3^5=243$; area ${\left(\frac{3}{4}\right)}^5=\frac{243}{1024}$. Rules: $t_n=3^n$, $t_0=1$, $t_n=3t_{n-1}$; $s_n={\left(\frac{3}{4}\right)}^n$, $s_0=1$, $s_n=\frac{3}{4}{s}_{n-1}$.

11. Q11 · 3 marks · Short answerHOTS / competency

    Find the smallest n for which $1+2+\cdots +n$ is more than 500.

    Hint

    $n(n+1)>1000$.

    Answer

    $S_{31}=496$ and $S_{32}=528$, so $n=32$.

12. Q12 · 3 marks · Short answerHOTS / competency

    You can climb a staircase 1 step or 2 steps at a time. Let $W_n$ be the number of ways to climb n steps, so $W_1=1$ and $W_2=2$. Explain why $W_n=W_{n-1}+W_{n-2}$ and find $W_6$.

    Hint

    Think about the last move.

    Answer

    The last move is either 1 step (after $W_{n-1}$ ways) or 2 steps (after $W_{n-2}$ ways), so the counts add. $W$: 1, 2, 3, 5, 8, 13; $W_6=13$ (the Virahānka–Fibonacci sequence).

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.What is the difference between an explicit and a recursive rule?Show answer

   An explicit rule gives $t_n$ directly from n; a recursive rule gives $t_n$ from earlier terms.

2. 2.Write the n-th term of an AP.Show answer

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

3. 3.Write the n-th term of a GP.Show answer

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

4. 4.What is $1+2+\cdots +n$?Show answer

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

5. 5.Who first wrote down the sequence 1, 2, 3, 5, 8, …, and in what context?Show answer

   Virahānka, in the 7th century CE, while studying the metres of Prakrit poetry.

6. 6.Which Indian text records the sum of the first n natural numbers?Show answer

   Āryabhaṭa's Āryabhaṭīya (Chapter 2, verse 19).

7. 7.How many black triangles are there at stage n of the Sierpiński triangle?Show answer

   $3^n$

8. 8.What happens to the black area of the Sierpiński triangle as n grows?Show answer

   It shrinks towards 0, as ${\left(\frac{3}{4}\right)}^n$.

9. 9.What is the least number of moves for n discs in the Tower of Hanoi?Show answer

   $2^n-1$

10. 10.On a graph, how do the points $(n,t_n)$ of an AP lie?Show answer

    On a straight line whose slope is d.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you write both rules for 5, 8, 11, 14, …?
• ☐Can you find which term of 21, 18, 15, … is −81?
• ☐Can you decide whether 0 is a term of that AP?
• ☐Can you find an AP from its 4th and 10th terms?
• ☐Can you prove $1+2+\cdots +n=\frac{n(n+1)}{2}$ by pairing?
• ☐Can you find $40+41+\cdots +80$?
• ☐Can you find the 8th term of the GP 5, 10, 20, …?
• ☐Can you find which term of 2, 6, 18, … is 4374?
• ☐Can you write the area of the Sierpiński carpet at stage n?
• ☐Can you explain why the Tower of Hanoi needs $2^n-1$ moves?

Sequences and Progressions: frequently asked questions