How a fixed common difference builds every term and sum
An arithmetic progression (AP) is a list of numbers in which each term is obtained by adding the same fixed number, the common difference d, to the term before it. In 3, 7, 11, 15, … the first term a is 3 and d is 4. The difference can be positive, negative or zero, so 20, 17, 14, … is also an AP with d = −3. The nth term is aₙ = a + (n − 1)d, which lets you jump straight to any term: the 50th term of 3, 7, 11, … is 3 + 49 × 4 = 199.
The sum of the first n terms is Sₙ = n/2 [2a + (n − 1)d], or Sₙ = n/2 (a + l) when the last term l is known. The second form comes from pairing the first and last terms, the second and second-last, and so on, since every pair has the same total. Adding 1 to 100 this way gives 50 pairs of 101, which is 5050. When a question gives two terms, write each as a + (n − 1)d and solve the two equations for a and d.
Why it matters
APs model anything that grows or shrinks by equal steps: savings that rise by a fixed amount each month, the number of seats in successive rows of an auditorium, or a loan repaid in instalments that fall by the same amount. In Class 11 the idea extends to geometric progressions and series, and the sum formula reappears in counting problems and in calculus.
Key terms
- Common difference
- The fixed amount added to each term of an AP to get the next term.
- nth term
- The general term aₙ = a + (n − 1)d that gives any position in the progression directly.
- Sum of n terms
- The total Sₙ of the first n terms, found as n/2 [2a + (n − 1)d].