Difference between AP and GP?

Chapter 5 of 14

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Gujarat (GSEB)Class 10 Mathematics★ 8-10 marks · CBSE Class 10 Maths (Chapter 5)

Arithmetic Progressions

Arithmetic Progressions (AP) — sequences with constant difference between consecutive terms. nth term formula, sum of n terms, applications in saving plans, loans, and patterns. CBSE Class 10 Maths Chapter 5.

⏱ 22 min readEasy difficulty✓ Free · GSEB syllabus

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By the end of this chapter you'll be able to…

1Identify AP from sequences
2Find nth term using aₙ = a + (n−1)d
3Calculate sum of n terms
4Solve word problems
5Use AP properties (three or five terms)

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Why this chapter matters

AP appears in finance (EMI, increments), physics, daily patterns. Foundation for Class 11 GP and infinite series.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Quadratic Equations

"Numbers in sequence — the rhythm of mathematics."

1. About the Chapter

Arithmetic Progression (AP) is a sequence where the DIFFERENCE between consecutive terms is constant. Used in:

Loans and EMIs
Salary increments
Saving plans
Physics (uniform motion)
Engineering and architecture

Definition

A sequence a₁, a₂, a₃, ... is an AP if aₙ − aₙ₋₁ = d (constant) for all n ≥ 2.

Components

a (first term): a₁ or just 'a'
d (common difference): aₙ − aₙ₋₁
n (number of terms)

Examples

2, 5, 8, 11, 14, ... (a = 2, d = 3)
10, 7, 4, 1, −2, ... (a = 10, d = −3)
100, 100, 100, ... (a = 100, d = 0)

2. The nth Term

Formula

aₙ = a + (n−1)d

where:

aₙ = nth term
a = first term
d = common difference
n = position

Derivation

a₁ = a
a₂ = a + d
a₃ = a + 2d
a₄ = a + 3d
...
aₙ = a + (n−1)d

Examples

Example 1: Find 20th term of 5, 8, 11, ...

a = 5, d = 3
a₂₀ = 5 + (20−1)(3) = 5 + 57 = 62

Example 2: Which term of 3, 8, 13, ... is 78?

a = 3, d = 5
aₙ = 3 + (n−1)(5) = 78
5n − 2 = 78 → n = 16

3. Sum of n Terms (Sₙ)

Formula 1 (when a, d, n known)

Sₙ = (n/2) [2a + (n−1)d]

Formula 2 (when a, l, n known, where l is the last term)

Sₙ = (n/2) (a + l)

Sum of First n Natural Numbers

1 + 2 + 3 + ... + n = n(n+1)/2 (where a = 1, d = 1, l = n)

Sum of First n Odd Numbers

1 + 3 + 5 + ... + (2n−1) = n² (a = 1, d = 2)

Sum of First n Even Numbers

2 + 4 + 6 + ... + 2n = n(n+1)

4. Properties of AP

Property 1

If you add (or subtract) same number to each term, it's still an AP with same d.

Property 2

If you multiply each term by k, it's still an AP with new d' = kd.

Property 3

Three terms in AP: (a−d), a, (a+d)

Property 4

Five terms in AP: (a−2d), (a−d), a, (a+d), (a+2d)

These properties simplify problem-solving.

5. Worked Examples

Example 1: Identify AP

Is 7, 13, 19, 25, ... an AP?

Differences: 6, 6, 6 — constant
YES, AP with a = 7, d = 6

Example 2: Find Specific Term

The 4th term of an AP is 14, 12th term is 38. Find first term and common difference.

a + 3d = 14 ... (i)
a + 11d = 38 ... (ii)
Subtract: 8d = 24 → d = 3
a = 14 − 9 = 5
AP: 5, 8, 11, 14, ...

Example 3: Sum

Find sum: 2 + 5 + 8 + ... + 32

a = 2, d = 3, l = 32
32 = 2 + (n−1)(3) → n = 11
S = (11/2)(2 + 32) = (11/2)(34) = 187

Example 4: Word Problem

A man saves ₹100 in first year, ₹200 in second, ₹300 in third, and so on. How much in 10 years?

a = 100, d = 100, n = 10
S = (10/2)[2(100) + 9(100)] = 5(1100) = ₹5500

Example 5: Three Terms in AP

Three numbers are in AP. Their sum = 24, product = 440. Find them.

Let numbers be (a−d), a, (a+d)
Sum: 3a = 24 → a = 8
Product: (8−d)(8)(8+d) = 440
8(64 − d²) = 440 → 64 − d² = 55 → d² = 9 → d = ±3
Numbers: 5, 8, 11 (or 11, 8, 5)

6. Common Mistakes

Confusing d with first term
- d is COMMON DIFFERENCE, not a₁.
Wrong formula for nth term
- aₙ = a + (n−1)d, not a + nd.
Sum formula confusion
- Sₙ = (n/2)[2a + (n−1)d] = (n/2)(a + l)
Negative d in increasing sequence
- If terms increase, d > 0; decrease, d < 0.
Not identifying AP
- Check differences are CONSTANT. Otherwise not AP.

7. Real-World Applications

Personal Finance

Salary increment: ₹3000/year increment is AP
EMI patterns: some loan structures use AP
Recurring deposits: monthly deposits

Engineering

Construction layers
Bridge cable lengths
Heat treatment temperatures

Sports/Statistics

Score progressions
Race timings

Physics

Uniform velocity: positions form AP
Falling object: positions every second form AP

8. Indian Heritage

Indian mathematicians worked on series:

Aryabhata (~5th century): sum formulas
Mahavira (~9th century): comprehensive treatment
Bhaskara II (12th century): general formulas
Madhava (14th century): infinite series

Gauss's 'add pairs from ends' trick is essentially what Indians had been using.

9. Conclusion

Arithmetic Progressions are SIMPLE but POWERFUL:

Recognise patterns
Find specific terms
Calculate sums efficiently
Apply to real life (finance, physics, engineering)

In Class 11, you'll learn GEOMETRIC PROGRESSIONS (multiply by constant ratio) and INFINITE SERIES. This chapter is the foundation.

Practice 20+ problems. Master formulas. Apply to word problems.

AP: The mathematical music of regularly spaced numbers.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

nth term

aₙ = a + (n−1)d

Sum formula 1

Sₙ = (n/2)[2a + (n−1)d]

When a, d, n given

Sum formula 2

Sₙ = (n/2)(a + l)

When a, l, n given

Sum of 1 to n

n(n+1)/2

Sum of first n odd

n²

Three terms in AP

(a−d), a, (a+d)

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ nth term wrong formula

✓ aₙ = a + (n−1)d. The (n−1) is key, not n.

WATCH OUT

✗ Confusing terms position

✓ Position n starts from 1. a₁ is first term, a₂ is second, etc.

WATCH OUT

✗ Negative d ignored

✓ AP can decrease (d < 0). Always check sign of d.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Practice problems

Work through this chapter's problems as a readiness check — reveal each solution, mark yourself honestly, and get your gap report at the end.

Readiness check

Are you exam-ready for Arithmetic Progressions?

4 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

4 questions~3 min worth ~10 marks in Gujarat (GSEB) exams

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•AP: sequence with constant difference d
•nth term: aₙ = a + (n−1)d
•Sum: Sₙ = (n/2)[2a + (n−1)d] = (n/2)(a + l)
•Sum of 1 to n: n(n+1)/2
•Sum of n odd: n²
•Three terms in AP: (a−d), a, (a+d)
•Common Indian heritage: Aryabhata, Bhaskara II

Gujarat (GSEB) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 8-10 marks per chapter

Question type	Marks each	Typical count	What it tests
MCQ	1	2	Identify AP, formulas
Short Answer	2-3	2	nth term, sum
Long Answer	5	1	Word problems, find unknown AP

Prep strategy

Memorise nth term and sum formulas
Practice 15+ problems
Word problems on saving, salary, scores
Master finding common difference

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Salary increments

Annual increment of fixed amount forms an AP.

EMI loans

Equal monthly instalments use AP-like structures.

Savings plans

Recurring deposits accumulate in AP pattern (with constant deposit).

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Write formula at start

Find a and d

Apply correct formula

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

STRETCH

Geometric Progressions

STRETCH

Infinite series

STRETCH

Telescoping series

STRETCH

Indian Madhava's series for π

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 10 BoardVery High

Maths OlympiadHigh

JEE FoundationVery High

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

AP (Arithmetic Progression): constant DIFFERENCE (add same number each time). GP (Geometric Progression): constant RATIO (multiply by same number each time). 2, 5, 8, 11 is AP (+3). 2, 6, 18, 54 is GP (×3). You'll study GP in Class 11.

Practice more, ask doubts, find a tutor

Gujarat (GSEB) Class 10 mock tests

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Related chapters

Students who read Arithmetic Progressions usually read these next.

Real Numbers

The Fundamental Theorem of Arithmetic, HCF and LCM through prime factorisation, and proofs that numbers like √5 are irrational.

Polynomials

Zeroes of a polynomial, what they mean on a graph, and the relationship between the zeroes and coefficients of a quadratic.

Pair of Linear Equations in Two Variables

Solving a pair of linear equations graphically and algebraically — by substitution and by elimination — plus consistency conditions and word problems. CBSE Class 10 Maths Chapter 3.

Quadratic Equations

Degree-2 equations ax² + bx + c = 0 — solving by factorisation and by the quadratic formula, the discriminant and the nature of roots, and word problems on area, speed, age and geometry. CBSE Class 10 Maths Chapter 4.

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