01 · KEY FORMULAS
Formulas for Sum of n Terms (Sₙ)
The sum of the first n terms of an AP with first term a and common difference d is given by:
Sₙ = (n / 2) [ 2a + (n - 1)d ]Sₙ = (n / 2) [ a + L ]Use the second formula when the last term L (or aₙ) is known!
Relationship Between aₙ and Sₙ
The n-th term of an AP can also be found from the sum of terms using:
aₙ = Sₙ - Sₙ₋₁
02 · SOLVED NCERT EXAMPLES
Step-by-Step Problem Solving
EXAMPLE 1
Find the sum of the first 22 terms of the AP: 8, 3, -2...
Given: a = 8, d = 3 - 8 = -5, n = 22.
Using Sₙ = (n/2) [2a + (n - 1)d]:
S₂₂ = (22 / 2) [ 2(8) + (22 - 1)(-5) ]
S₂₂ = 11 [ 16 + 21(-5) ]
S₂₂ = 11 [ 16 - 105 ] = 11 × (-89) = -979
S₂₂ = 11 [ 16 + 21(-5) ]
S₂₂ = 11 [ 16 - 105 ] = 11 × (-89) = -979
EXAMPLE 2 (NCERT WORD PROBLEM)
Find the sum of first 1000 positive integers
Let S = 1 + 2 + 3 + ... + 1000. Here, a = 1, L = 1000, n = 1000.
Using Sₙ = (n/2) [a + L]:
S₁₀₀₀ = (1000 / 2) [ 1 + 1000 ]
S₁₀₀₀ = 500 × 1001 = 500500
S₁₀₀₀ = 500 × 1001 = 500500
General Formula for sum of first n positive integers: Sₙ = n(n + 1) / 2.
03 · QUICK SUMMARY
Key Takeaways
01
Standard sum formula: Sₙ = (n/2)[2a + (n-1)d].
02
Short formula when last term L is given: Sₙ = (n/2)[a + L].
03
Finding a term from sum: aₙ = Sₙ - Sₙ₋₁.