Class 9 Maths · Original EZT practice
Sequences and Progressions (Class 9 Chapter 8): summary and practice
A sequence is numbers following a rule - and once you find the rule, you can predict any term without listing them all. This chapter builds that step by step: explicit and recursive rules, arithmetic progressions, the famous shortcut for adding the first n natural numbers, and geometric progressions. All questions are original EZT tutor practice following the chapter's own section order.
6 original worked examples, 6 fresh self-check questions and 4 mixed questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 22 October 2026:
1. Explicit and recursive rules
An explicit rule gives the nth term straight from n, like tn = 1 + (n - 1) x 4. A recursive rule gives each term from the previous one plus a starting term, like tn = t(n-1) + 5 with t1 = 2.
2. Arithmetic progressions
An AP adds the same common difference d every step: a, a + d, a + 2d, ... So the nth term is tn = a + (n - 1)d - predict the 10th term without writing nine.
3. Sum of the first n natural numbers
Pairing first with last gives the trick credited to the young Gauss: 1 + 2 + ... + n = n(n + 1)/2. So 1 to 100 is 100 x 101 / 2 = 5050.
4. Geometric progressions
A GP multiplies by the same common ratio r every step: a, ar, ar^2, ... So tn = a x r^(n-1) - doubling from 1 gives 32 at the 6th term.
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
Use an explicit rule
Choose the method: Substitute n directly into the formula.
EZT practice question
A sequence has the explicit rule tn = 1 + (n - 1) x 4. Find the 5th term.
Show the worked solution
- An explicit rule gives the term straight from n.
- t5 = 1 + (5 - 1) x 4.
- That is 1 + 16 = 17.
Answer: 17.
Check: List it out: 1, 5, 9, 13, 17 - the 5th term is indeed 17, no listing needed with the rule.
Watch for this: Use (n - 1), not n - the first term already sits at step 1.
Now try it yourself
Find the 6th term of tn = 3n - 1.
Check your answer
17.
3 x 6 - 1 = 17.
Write a recursive rule
Choose the method: Each term comes from the previous one, plus a starting term.
EZT practice question
A sequence starts at 2 and each term is 5 more than the previous one. Write the first four terms and the recursive rule.
Show the worked solution
- Start: t1 = 2.
- Add 5 each step: 2, 7, 12, 17.
- Rule: tn = t(n-1) + 5 with t1 = 2.
Answer: 2, 7, 12, 17; tn = t(n-1) + 5 with t1 = 2.
Check: Each term minus the previous is 5 - the recursion captures exactly that, and the starting term pins it down.
Watch for this: A recursive rule is incomplete without the starting term - '+5' alone fits infinitely many sequences.
Now try it yourself
Next term after 17 in this sequence?
Check your answer
22.
17 + 5 = 22.
Find the nth term of an AP
Choose the method: tn = a + (n - 1)d.
EZT practice question
Find the 10th term of the AP 5, 9, 13, 17, ...
Show the worked solution
- First term a = 5, common difference d = 9 - 5 = 4.
- t10 = 5 + (10 - 1) x 4.
- That is 5 + 36 = 41.
Answer: 41.
Check: The general form a, a + d, a + 2d, ... places the nth term at a + (n - 1)d - nine jumps of 4 past the first term.
Watch for this: Is 11, 7, 3, -1, ... an AP? Yes - d can be negative (here d = -4).
Now try it yourself
Taxi pattern 240, 280, 320, ... per extra kilometre - nth term?
Check your answer
tn = 200 + 40n.
240 + (n - 1) x 40 simplifies to that.
Sum the first n natural numbers
Choose the method: Pair first with last: n(n + 1)/2.
EZT practice question
Find the sum 1 + 2 + 3 + ... + 100.
Show the worked solution
- Pair 1 with 100, 2 with 99 - every pair sums to 101.
- There are 50 such pairs.
- S = 100 x 101 / 2 = 5050.
Answer: 5050.
Check: The pairing argument is the formula n(n + 1)/2 - the trick credited to the young Gauss.
Watch for this: Do not add term by term - the pairing formula is the point of the section.
Now try it yourself
Find 1 + 2 + ... + 50.
Check your answer
1275.
50 x 51 / 2 = 1275.
Work the sum backwards
Choose the method: Set n(n + 1)/2 equal to the target and solve.
EZT practice question
The sum of the first n natural numbers is 210. Find n.
Show the worked solution
- n(n + 1)/2 = 210, so n(n + 1) = 420.
- Look for consecutive numbers with product 420.
- 20 x 21 = 420 - so n = 20.
Answer: n = 20.
Check: Verify: 20 x 21 / 2 = 210 - the sum formula run in reverse.
Watch for this: n(n + 1) = 420 is not n^2 = 420 - the two factors differ by 1.
Now try it yourself
Sum of the first n natural numbers is 55. Find n.
Check your answer
n = 10.
10 x 11 = 110 = 2 x 55.
Find the nth term of a GP
Choose the method: tn = a x r^(n-1).
EZT practice question
Is 1, 2, 4, 8, 16, ... a geometric progression? If so, find its 6th term.
Show the worked solution
- Each term divided by the previous gives 2 - a constant ratio, so yes, a GP.
- a = 1 and r = 2.
- t6 = 1 x 2^5 = 32.
Answer: Yes, with r = 2; the 6th term is 32.
Check: In a GP you multiply by r each step: a, ar, ar^2, ... - the nth term carries r^(n-1), one fewer multiplication than the term number.
Watch for this: AP adds, GP multiplies - check the RATIO of terms, not the difference.
Now try it yourself
A fractal stage has 3 times the triangles of the previous stage, starting from 1. Stage 5?
Check your answer
81.
1 x 3^4 = 81.
Try a mixed set
Choose a method or explanation before opening each answer.
1. Explicit rule?
Show answer and reason
Gives the nth term directly from n.
Like tn = 3n - 1.
2. Recursive rule?
Show answer and reason
Each term from the previous, plus a start.
Like tn = t(n-1) + 5 with t1 = 2.
3. AP nth term?
Show answer and reason
tn = a + (n - 1)d.
Common difference d added each step.
4. GP nth term?
Show answer and reason
tn = a x r^(n-1).
Common ratio r multiplied each step.
Try it yourself
Work these on paper before checking any notes. Answers are deliberately not shown here.
- Find the 8th term of the AP 3, 10, 17, ...
- Find 1 + 2 + ... + 60.
- Find the 5th term of the GP 2, 6, 18, ...
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Questions about this practice page
- What is an arithmetic progression in Class 9?
- A sequence where each term is the previous term plus a fixed common difference d, like 1, 4, 7, 10 with d = 3. Its nth term is tn = a + (n - 1)d.
- What is a geometric progression?
- A sequence where each term is the previous term times a fixed common ratio r, like 1, 2, 4, 8 with r = 2. Its nth term is tn = a x r^(n-1).
- What is the sum of the first n natural numbers?
- n(n + 1)/2 - for example, 1 to 100 adds to 5050. The trick pairs the first and last terms, and is credited to the young Gauss.
- What is the difference between an explicit and a recursive rule?
- An explicit rule computes any term directly from n. A recursive rule defines each term using the previous term, so it also needs the starting term.
- Are these the NCERT exercise questions for Class 9 Maths Chapter 8?
- No. EZT pages are original practice following the chapter's section order - use them to prepare for the textbook exercises, not as a substitute for your book.