Class 10 Arithmetic Progressions Complete Guide 2027 — nth Term, Sum, Word Problems + Tricks

In short: Arithmetic Progressions (AP) carries 7-8 marks in CBSE Class 10 Maths 2027 and is one of the most formula-driven chapters on the paper. Two formulas do all the heavy lifting — aₙ = a + (n−1)d for the nth term and Sₙ = n/2[2a + (n−1)d] for the sum. Learn them with the tricks below, master the five board exam question types, and follow the practice plan to lock in these marks.

Arithmetic Progressions is Chapter 5 in CBSE Class 10 Maths and it falls under the Algebra unit, which carries a total of 20 marks — the highest-weighted unit on the paper. AP alone accounts for 7-8 of those marks. It is a chapter that rewards practice over concept depth: the formulas are few but the question variety is wide. Word problems on AP are a board exam favourite and appear almost every year.

If you have already worked through Quadratic Equations and Coordinate Geometry, the algebra in AP will feel familiar. For the complete marks breakdown, see the CBSE Class 10 chapter-wise weightage.

What Is an Arithmetic Progression?

An Arithmetic Progression is a sequence of numbers where the difference between any two consecutive terms is always the same. This constant difference is called the common difference (d).

Examples:

• 2, 5, 8, 11, 14, ... (a = 2, d = 3)
• 100, 90, 80, 70, ... (a = 100, d = −10)
• 3, 3, 3, 3, ... (a = 3, d = 0) — yes, a constant sequence is an AP

Key terms:

• a = first term
• d = common difference = any term minus the previous term
• n = number of terms
• aₙ (or l) = nth term (last term)
• Sₙ = sum of first n terms

To check if a sequence is an AP: verify that a₂ − a₁ = a₃ − a₂ = a₄ − a₃. If the differences are all equal, it is an AP.

The Essential Formulas

Formula 1: nth Term of an AP

aₙ = a + (n − 1)d

This finds any term in the sequence if you know the first term and common difference.

Memory trick — "First term plus the gap, times one less than the position": You start at a and take (n−1) steps of size d. Think of it as climbing stairs: if you are on the first step (a), to reach the nth step, you climb (n−1) more steps.

Variations you must know:

• Finding d when two terms are known: d = (aₙ − a)/(n − 1)
• Finding n (number of terms): n = (aₙ − a)/d + 1
• The term from the end: the kth term from the end = l − (k−1)d, where l is the last term

Formula 2: Sum of First n Terms

Sₙ = n/2 [2a + (n − 1)d]

This finds the sum of the first n terms of an AP.

Memory trick — "n over 2, times the bookends": The formula is essentially n/2 × (first term + last term). Since the last term = a + (n−1)d, the full formula expands to n/2[2a + (n−1)d]. Think of it as: "average of first and last term, multiplied by the number of terms."

Alternative form (when last term is known):

Sₙ = n/2 (a + l)

This is simpler and faster when you know the last term. Board questions often give you the last term directly or ask you to find it first.

Key relationship — connecting Sₙ and aₙ:

aₙ = Sₙ − Sₙ₋₁ (for n ≥ 2)

This is how you find an individual term when you are given the sum formula. Board exams love this.

Formula 3: Sum of First n Natural Numbers

1 + 2 + 3 + ... + n = n(n+1)/2

This is a special case of the AP sum formula with a = 1, d = 1. It appears in word problems frequently.

How Board Exam Questions Actually Look

AP questions follow five standard patterns. Recognise them and your solving approach becomes automatic.

Type 1: Find the nth Term (1-2 marks)

"Find the 20th term of the AP: 2, 7, 12, 17, ..."

1. Identify: a = 2, d = 7 − 2 = 5, n = 20
2. aₙ = a + (n−1)d = 2 + 19 × 5 = 2 + 95 = 97

Type 2: Find the Number of Terms (2-3 marks)

"How many terms of the AP 24, 21, 18, ... must be taken so that the sum is 78?"

1. Identify: a = 24, d = −3, Sₙ = 78
2. Use Sₙ = n/2[2a + (n−1)d]: 78 = n/2[48 + (n−1)(−3)]
3. 156 = n[48 − 3n + 3] = n[51 − 3n]
4. 3n² − 51n + 156 = 0 → n² − 17n + 52 = 0
5. (n − 4)(n − 13) = 0, so n = 4 or n = 13

Both answers are valid. When the AP has negative d, the sum can reach the same value twice (once while increasing, once while the negative terms start subtracting). State both answers.

Type 3: Find the Sum (2-3 marks)

"Find the sum of first 15 terms of the AP: 3, 7, 11, ..."

1. a = 3, d = 4, n = 15
2. Sₙ = 15/2[2(3) + 14(4)] = 15/2[6 + 56] = 15/2 × 62 = 465

Type 4: Finding a Term from Sₙ (3 marks)

"The sum of first n terms of an AP is given by Sₙ = 3n² + 5n. Find the AP and its 15th term."

1. a₁ = S₁ = 3(1) + 5(1) = 8 (first term)
2. a₂ = S₂ − S₁ = [3(4) + 10] − 8 = 22 − 8 = 14
3. d = 14 − 8 = 6
4. a₁₅ = 8 + 14 × 6 = 8 + 84 = 92
5. AP: 8, 14, 20, 26, ...

Type 5: Word Problems (4-5 marks)

"A sum of ₹700 is to be distributed among some people in such a way that each person gets ₹10 more than the previous person. If the first person gets ₹40, how many people are there?"

1. This is an AP: a = 40, d = 10, Sₙ = 700
2. 700 = n/2[2(40) + (n−1)(10)]
3. 1400 = n[80 + 10n − 10] = n[70 + 10n]
4. 10n² + 70n − 1400 = 0 → n² + 7n − 140 = 0
5. (n + 14)(n − 10) = 0, so n = 10 (rejecting negative value)
6. Answer: 10 people

Common word problem themes: Money distribution, saving schemes (saving ₹x more each month), counting objects arranged in patterns (rows of seats, stacked logs), and age-based problems.

Common Mistakes That Cost Marks

1. Confusing aₙ and Sₙ formulas. The nth term formula gives ONE term. The sum formula gives the TOTAL of all terms. If the question asks "find the 10th term," use aₙ. If it asks "find the sum of 10 terms," use Sₙ. Reading the question carefully saves you from this error.

2. Getting d wrong when the AP is decreasing. In the AP 20, 17, 14, 11, ..., d = 17 − 20 = −3 (negative). Students often write d = 3 and get the wrong answer. Always calculate d as (second term − first term), preserving the sign.

3. Using n instead of (n−1) in the nth term formula. The formula is a + (n−1)d, NOT a + nd. The (n−1) is there because the first term is already counted. This is the single most common substitution error in AP questions.

4. Forgetting to check if n is a positive integer. When solving for n, you might get a fractional or negative answer. The number of terms must be a positive integer. If n comes out as 7.5, it means the given sum cannot be achieved with a whole number of terms — check your calculation.

5. Not forming the AP before solving word problems. Word problems give you a story. Convert it to an AP by identifying a, d, and what is being asked (aₙ or Sₙ) before touching the formula. Writing "a = ..., d = ..., need to find ..." at the start of your solution earns method marks.

4-Week Practice Plan

Week 1: Basics — Identifying APs and Finding Terms

• Read NCERT Section 5.1 and 5.2
• Solve all questions in Exercise 5.1 (identifying APs, finding common difference)
• Solve Exercise 5.2 (finding nth term, number of terms)
• Daily drill: write the nth term formula from memory, then solve 3 "find the nth term" problems

Week 2: Sum of Terms

• Read NCERT Section 5.3
• Solve all questions in Exercise 5.3 (sum of n terms, both formula forms)
• Practice the aₙ = Sₙ − Sₙ₋₁ type — solve at least 5 such problems
• Practice the "how many terms for a given sum" type — these often produce quadratic equations, so your Quadratic Equations skills come in handy

Week 3: Word Problems

• Solve Exercise 5.4 completely (word problems)
• Categorise problems by type: money, counting, age, pattern
• Solve 5 additional word problems from CBSE important questions
• For each word problem, practice the method: read → identify a, d → identify what to find → choose formula → solve → verify

Week 4: Board Paper Practice and Speed

• Solve AP questions from the last 5 years of CBSE board papers. See how to use previous year papers for strategy.
• Time yourself: 2-mark questions in 3 minutes, 4-mark questions in 6 minutes
• Solve the CBSE 2027 sample paper AP questions
• Final speed round: 6 mixed AP problems in 30 minutes

Quick Revision Card for Exam Day

• nth term: aₙ = a + (n−1)d → "Start plus steps times gap"
• Sum: Sₙ = n/2[2a + (n−1)d] → "Half of n, times double start plus steps times gap"
• Sum (if last term known): Sₙ = n/2(a + l) → "Half of n, times first plus last"
• Term from sum: aₙ = Sₙ − Sₙ₋₁
• Number of terms: n = (l − a)/d + 1
• Sum of 1 to n: n(n+1)/2
• d can be NEGATIVE — always check the sign
• n must be a POSITIVE INTEGER — if it is not, recheck your work
• For word problems: identify a, d, and whether you need aₙ or Sₙ BEFORE substituting

For guided practice on every AP question type with step-by-step hints, try Padhaao's AI tutor — it adapts to your pace and explains where you went wrong.

Frequently Asked Questions

How many marks does Arithmetic Progressions carry in CBSE Class 10 Maths 2027?

AP carries 7-8 marks in the CBSE Class 10 board exam 2027. It is part of the Algebra unit (20 marks total). You can expect one 2-3 mark question on finding terms or sums, and one 4-5 mark word problem. Some years also include a 1-mark MCQ on identifying an AP or finding the common difference.

What is the difference between the nth term formula and the sum formula?

The nth term formula (aₙ = a + (n−1)d) gives you the value of a single specific term in the AP. The sum formula (Sₙ = n/2[2a + (n−1)d]) gives you the total of all terms from the first to the nth. If the question says "find the 10th term," use aₙ. If it says "find the sum of the first 10 terms," use Sₙ.

How do I solve AP word problems in CBSE board exams?

First, read the problem and identify the AP: what is the first term (a), what is the common difference (d), and what quantity is being asked for. Then decide if you need the nth term (aₙ) or the sum (Sₙ). Write "a = ..., d = ..., find ..." before substituting into the formula. Common themes include money distribution, savings schemes, and counting patterns. The algebra often leads to a quadratic equation, so solving quadratics is a prerequisite skill.

Can the common difference of an AP be zero or negative?

Yes. If d = 0, every term is the same (e.g., 5, 5, 5, 5, ...) — this is still a valid AP. If d is negative, the AP is decreasing (e.g., 20, 17, 14, 11, ...). Always calculate d as (second term − first term) and preserve the sign. A negative d is a very common source of errors because students accidentally use the positive value.

What is the relationship between Sₙ and aₙ?

The nth term can be found from the sum formula using aₙ = Sₙ − Sₙ₋₁ for n ≥ 2, and a₁ = S₁. This means if you know the sum formula (for example, Sₙ = 3n² + 5n), you can find any individual term by computing consecutive sums and subtracting. Board exams frequently test this relationship, so practise it until it is automatic.

Is NCERT enough for AP in CBSE board exams?

Yes, NCERT is sufficient if you solve all exercises thoroughly, including the optional Exercise 5.4. The board exam questions closely follow NCERT patterns. After completing NCERT, solve 3-5 years of previous year board questions for this chapter to see the exact difficulty level and time yourself. If you can solve every NCERT problem and every past paper question confidently, you are fully prepared.

Arithmetic Progressions is a formula-driven chapter with predictable patterns. Two formulas, five question types, and consistent practice — that is the recipe for full marks.