Sequence & Series - How It Appears in JEE

6 recurring patterns. Learn the pattern, recognize it in 5 seconds, apply the right approach.

Pattern

Given conditions on terms or sums, find a specific term or sum

How to recognize

Question involves S_n, a_n, common difference d, or ratio of terms in an AP

→ Set up equations using a_n = a+(n-1)d and S_n = n/2(2a+(n-1)d). Solve the system.

Pattern

Find terms, ratio, or sum of a GP given constraints

How to recognize

Mentions geometric progression, common ratio, or product/sum conditions on GP terms

→ Use a_n = ar^(n-1). For three terms, take them as a/r, a, ar. Product conditions simplify nicely.

Pattern

Terms form both AP and GP under different conditions

How to recognize

Some terms in AP, some (modified) terms in GP, or AP-GP relationship

→ Write AP terms using a,d. Substitute into GP condition (b^2=ac for three terms). Solve simultaneously.

Pattern

Sum series by partial fractions that cancel adjacent terms

How to recognize

General term has products in denominator like 1/(n(n+1)) or 1/((2n-1)(2n+1))

→ Decompose into partial fractions. Most terms cancel, leaving only first and last.

Pattern

Sum a series whose terms don't form a standard AP or GP

How to recognize

Series like 5+11+19+29+... where differences form an AP or GP

→ Find T_n using S_n - S_{n-1}. If first differences are AP, T_n is quadratic. Sum using standard formulas.

Pattern

Find sum of infinite GP or use convergence conditions

How to recognize

Infinite series, |r|<1, sum to infinity, or recurring decimals

→ Use S = a/(1-r). For sum of powers: sum of a^k*r^(kn) = a^k/(1-r^k).