Arithmetic Progressions — Class 10 Mathematics
"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.
