Concept Topic GuideCBSE & ICSEClass 10MathematicsArithmetic ProgressionsVerified Faculty Notes & Solutions
Arithmetic Progressions (AP) – Class 10 Maths Chapter 5 Formulas & Notes
Class 10 AP notes: nth term formula, sum of n terms, common difference, solved examples for arithmetic progressions CBSE & ICSE.
7 students readUpdated 4 September 2026
Chapter 5: Arithmetic Progressions (AP)
Definition: An Arithmetic Progression is a sequence of numbers in which each term after the first is obtained by adding a fixed number called the common difference (d) to the preceding term.
General form: a, a+d, a+2d, a+3d, ... where a = first term, d = common difference.
General form: a, a+d, a+2d, a+3d, ... where a = first term, d = common difference.
5.1 Common Difference
d = a₂ − a₁ = a₃ − a₂ = aₙ − aₙ₋₁ (constant for all consecutive terms)
- If d > 0: AP is increasing (e.g., 2, 5, 8, 11, ...)
- If d < 0: AP is decreasing (e.g., 10, 7, 4, 1, ...)
- If d = 0: All terms are equal (e.g., 5, 5, 5, 5, ...)
5.2 nth Term of an AP
aₙ = a + (n − 1)d
Where: aₙ = nth term, a = first term, n = term number, d = common difference
Where: aₙ = nth term, a = first term, n = term number, d = common difference
Example: Find the 20th term of the AP: 3, 7, 11, 15, ...
Here a = 3, d = 7 − 3 = 4, n = 20
a₂₀ = 3 + (20 − 1) × 4 = 3 + 76 = 79
5.3 Sum of First n Terms
Sₙ = n/2 × [2a + (n − 1)d]
OR (if last term l is known):
Sₙ = n/2 × (a + l)
Where: Sₙ = sum of first n terms, a = first term, l = last term, d = common difference
OR (if last term l is known):
Sₙ = n/2 × (a + l)
Where: Sₙ = sum of first n terms, a = first term, l = last term, d = common difference
Example: Find the sum of the first 15 terms of the AP: 2, 7, 12, 17, ...
a = 2, d = 5, n = 15
S₁₅ = 15/2 × [2(2) + (15 − 1)(5)] = 15/2 × [4 + 70] = 15/2 × 74 = 555
5.4 Important Properties
- The nth term can also be found using: aₙ = Sₙ − Sₙ₋₁
- If three numbers a, b, c are in AP, then 2b = a + c (i.e., the middle term is the arithmetic mean of the other two)
- Sum of first n natural numbers: Sₙ = n(n+1)/2
Board Exam Tip: Many AP problems can be simplified by choosing terms wisely:
• Three terms in AP: take as (a−d), a, (a+d)
• Four terms in AP: take as (a−3d), (a−d), (a+d), (a+3d) (common difference = 2d)
This eliminates one variable and simplifies the algebra significantly.
• Three terms in AP: take as (a−d), a, (a+d)
• Four terms in AP: take as (a−3d), (a−d), (a+d), (a+3d) (common difference = 2d)
This eliminates one variable and simplifies the algebra significantly.
Indexed Topics & Examination Keywords
#arithmetic progression class 10#AP formula#nth term#sum of n terms#common difference#NCERT maths#CBSE#ICSE#AP problems
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