Master the formula derivation and solve real-world problems
nth Term Formula
Learn to find any term in an arithmetic progression using the powerful nth term formula.
We'll derive the formula step-by-step and apply it to real problems.
aₙ = a + (n-1)d
Formula Derivation
Step-by-Step Development
Let's derive the nth term formula using Reena's salary progression:
Year 1
₹8,000
Year 2
₹8,500
Year 3
₹9,000
Year 4
₹9,500
1st term: a₁ = a = 8000
2nd term: a₂ = a + d = 8000 + 500
3rd term: a₃ = a + 2d = 8000 + 2×500
4th term: a₄ = a + 3d = 8000 + 3×500
Pattern: aₙ = a + (n-1)d
Worked Examples
Example 1: Find 10th term
AP: 2, 7, 12, ...
Given: a = 2, d = 5, n = 10
a₁₀ = 2 + (10-1)×5
a₁₀ = 2 + 9×5 = 47
Example 2: Find which term
AP: 21, 18, 15, ... which term is -81?
Given: a = 21, d = -3, aₙ = -81
-81 = 21 + (n-1)(-3)
-81 = 21 - 3n + 3
n = 35
Example 3: Check if term exists
Is 301 a term of 5, 11, 17, 23, ...?
Given: a = 5, d = 6, aₙ = 301
301 = 5 + (n-1)×6
n = 296/6 = 49.33...
Not integer → 301 is NOT a term
Interactive Solver
Find nth Term or Position
Enter the known values to solve for the unknown:
Real-World Applications
Finding Large Terms
100th term of 3, 8, 13, 18, ...
a = 3, d = 5
a₁₀₀ = 3 + (100-1)×5
a₁₀₀ = 3 + 495 = 498
Two-digit numbers ÷ 3
How many: 12, 15, 18, ..., 99?
a = 12, d = 3, last = 99
99 = 12 + (n-1)×3
n = 30 terms
Terms from End
5th term from end in 10, 7, 4, ..., -62
Total terms: n = 25
5th from end = 21st term
a₂₁ = 10 + (21-1)×(-3) = -50
Explore nth Term Concepts
Learning Progress
Master the nth term formula through step-by-step derivation, worked examples, and real-world applications.
Use the interactive solver to practice and verify your understanding.