This page is the routing map for ncert solutions for class 10 maths chapter 5, Arithmetic Progressions. It will not solve a single textbook question — every worked solution sits one level down on its own exercise page, inside the class 10 maths solutions hub.
What this page does instead is tell you which of the four exercises (5.1, 5.2, 5.3 or optional 5.4) your homework question sits in and which method that exercise wants you to use.
The chapter runs on three movements: recognising an AP, finding its nth term, and summing its first n terms. Each movement has its own exercise, so match your question to a movement and you know exactly where to go. The directory below, the formula map and the routing list in section four should get you there in under a minute.
| Exercise | What you get |
|---|---|
| Exercise 5.1 | Solved |
| Exercise 5.2 | Solved |
| Exercise 5.3 | Solved |
| Exercise 5.4 | Solved |
Exercise guide: what each exercise tests
The directory below lists every exercise in the chapter, its question count and the method it drills. The descriptions after it are drawn from each exercise’s own questions, so you can match your homework line by line.
- Exercise 5.1 — spot the AP (4 questions). The definition-only exercise. Q1 puts the AP test into four real situations — a taxi fare rising ₹8 per km, a vacuum pump pulling a quarter of the remaining air, well digging at ₹50 more per metre, and compound interest at 8% — and asks which list actually forms an AP. Q2 and Q3 run the test the other way: given the first term \(a\) and common difference \(d\), write four terms; or read \(a\) and \(d\) back off a list. Q4 is the 15-part classifier over squares, square roots, powers and the rabbit-pair list 1, 1, 2, 3, 5. Every part wants the single common-difference test \(a_{k+1} – a_k\).
- Exercise 5.2 — the nth term (20 questions). The whole exercise drills \(a_n = a + (n-1)d\). It opens with a five-row fill-in table hunting any one of \(a\), \(d\), \(n\), \(a_n\), then two MCQs and boxed missing terms. Next come the “which term equals…?” family, a term-from-the-last on 3, 8, 13, …, 253, and divisibility counts — three-digit multiples of 7 and multiples of 4 between 10 and 250. It closes with salary and weekly-savings word problems. Several late questions need two nth-term equations solved as a pair.
- Exercise 5.3 — the sums (20 questions). Everything here uses \(S = \frac{n}{2}[2a + (n-1)d]\) or its last-term twin. It opens with direct sums to a stated number of terms, then the ten-part data hunt where any three of \(a, d, n, S, a_n, l\) are given and you solve for the rest. One sum question admits two values of \(n\). Divisibility sums appear again (first 40 multiples of 6, first 15 multiples of 8, odd numbers between 0 and 50), followed by the word-problem run: a delay penalty, cash prizes, class-wise tree planting, a spiral of semicircles (Fig. 5.4), a log stack (Fig. 5.5) and a potato race (Fig. 5.6).
- Exercise 5.4 — optional (5 questions, p.23). The hardest material in the chapter, and NCERT explicitly marks it “not from the examination point of view”. It mixes the first negative term of 121, 117, 113, …, a sum-and-product pair driving \(S_{16}\), ladder rungs tapering 45 cm to 25 cm, splitting the houses numbered 1 to 49 at a house \(x\), and the concrete volume of 15 terrace steps.
The method map: every formula the chapter runs on
These are the only results the four exercises ever call on. Recognising which one your question mentions does most of the routing for you.
| Result | Formula | Where it appears |
|---|---|---|
| General form of an AP | \(a, a+d, a+2d, a+3d, \dots\) | Section 5.2 (p.3) |
| Common difference | \(d = a_{k+1} – a_k\) | Section 5.2 (p.3), drilled by 5.1 |
| nth term | \(a_n = a + (n-1)d\) | Section 5.3 (p.8), drilled by 5.2 |
| Sum of first n terms | \(S = \frac{n}{2}[2a + (n-1)d]\) | Section 5.4 (p.15), drilled by 5.3 |
| Sum from first and last term | \(S = \frac{n}{2}(a + l)\) | Section 5.4 (p.15) — the form to use when \(d\) is not given |
| Term from two sums | \(a_n = S_n – S_{n-1}\) | Section 5.4 remark (p.15) |
| Arithmetic mean | \(b = \frac{a + c}{2}\) when \(a, b, c\) are in AP | Note to the reader (p.24) |
So the chapter’s shape is simple: 5.1 drills the common-difference test, 5.2 drills the nth term, 5.3 drills the sums. You can verify every entry above against the official rationalised NCERT textbook PDF for Chapter 5 (Arithmetic Progressions).
Common mistakes that cost marks in AP problems
- The sign rule for the common difference. To find \(d\) you subtract the kth term from the (k+1)th term, even when the later term is smaller. The book’s own list 6, 3, 0, −3 gives \(d = -3\), not +3 (p.3).
- The off-by-one in a term from the last. The 11th term from the last of a 25-term AP is the 15th term, not the 14th. The textbook example 10, 7, 4, …, −62 makes exactly this point (p.8).
- Two answers for the number of terms. When \(a\) is positive and \(d\) is negative, later terms cancel earlier ones: in Example 13 the same sum 78 comes from both \(n = 4\) and \(n = 13\), and both values are admissible (p.15).
- Not every patterned list is an AP. 1, 1, 2, 3, 5 and lists of squares or powers fail the common-difference test, which is why 5.1 exists (p.3, p.7).
- \(a_n\) is a term, \(S_n\) is a sum. The remark \(a_n = S_n – S_{n-1}\) is the bridge that connects a single term to two sums (p.15).
How to pick your exercise in one glance
- “Is this list an AP?” or “why is / isn’t this an AP?” → Exercise 5.1 (questions 1 and 4 are the classifiers).
- “Write the first four terms” or “find the common difference” → Exercise 5.1 (questions 2 and 3).
- “Which term is…?”, “find the nth term”, “how many terms…?”, “divisible by 7 (or 4)”, “term from the last” → Exercise 5.2.
- “Find the sum”, “given \(a, d, \dots\) find \(S_n\)”, “sum of multiples of…” → Exercise 5.3.
- “First negative term”, “ladder rungs”, “houses 1 to 49”, “terrace steps” → Exercise 5.4 (optional, not from the examination point of view).
The full worked solutions for each exercise sit one level down on their own pages. This page sits inside the class 10 hub, and its neighbours — Chapter 4, Quadratic Equations, and Chapter 6, Triangles — have the same structure; if you arrived from a search engine, start from the site’s notes index.
| Exercise | What you get |
|---|---|
| Exercise 5.1 | Solved |
| Exercise 5.2 | Solved |
| Exercise 5.3 | Solved |
| Exercise 5.4 | Solved |
Arithmetic Progressions: frequently asked questions
Which exercise has the questions that ask whether a list is an AP?
Exercise 5.1 (4 questions). Q1 tests the AP test in four real situations and Q4 runs a 15-part classifier across squares, roots, powers and the list 1, 1, 2, 3, 5. Both use only the common-difference test.
Is Exercise 5.4 part of the CBSE board exam?
NCERT labels it “not from the examination point of view” (p.23). Treat it as enrichment — do it if you want the hardest AP questions, but board papers draw from 5.1 to 5.3.
How do I find the nth term of an AP?
Use \(a_n = a + (n-1)d\), where \(a\) is the first term and \(d\) the common difference (p.8). That single formula is the whole of Exercise 5.2.
When should I use \(S = \frac{n}{2}(a + l)\) instead of \(S = \frac{n}{2}[2a + (n-1)d]\)?
Use the \((a + l)\) form when the first and last terms are given and the common difference is not (p.15). Use the \(2a\) form when you know \(a\) and \(d\). Both give the same sum.
How do I find a term from the last of an AP?
First find the total number of terms, then count back. In a 25-term AP the 11th term from the last is the \((25 – 11 + 1) = 15\)th term, not the 14th (p.8).
Why can a sum question give two answers for the number of terms?
When the first term is positive and \(d\) is negative, later terms become negative and cancel earlier ones. Example 13 shows the sum 78 coming from both \(n = 4\) and \(n = 13\); both answers are admissible (p.15).
Reference: NCERT textbooks (CBSE).