Arithmetic Sequences: nth Term, Counting Terms and Sums
What the exam tests
Arithmetic Sequences questions ask you to recognise a constant difference, form the nth term, count terms and add a sequence. Kerala SSLC also uses a sequence given by an algebraic expression such as 4n+1, where the coefficient of n is the common difference.
Core rules
- Common difference d = x₂ − x₁; it may be negative or fractional.
- nth term: xₙ = x₁ + (n−1)d.
- Sum of n terms: Sₙ = n/2[2x₁ + (n−1)d] = n/2(x₁ + xₙ).
- If a sequence is an+b, it is arithmetic with d=a and first term a+b.
- To test whether a number belongs, solve x₁+(n−1)d = number for a positive integer n.
Worked example
For 7, 12, 17, …, d=5. The 20th term is 7+(20−1)×5=7+95=102. The sum of the first 20 terms is 20/2(7+102)=10×109=1090. Counting terms from 7 to 102 gives (102−7)/5+1=20, which checks n.
Traps
Using n instead of n−1 is the most common slip. In the sum formula, average the first and last terms only when the last term is the nth term. A sequence of squares is not arithmetic. Negative d reduces terms, so do not force d positive. For “how many terms” questions, n must be a whole number; otherwise the value is not a term.