Sequences and Series is an important chapter for all competitive exams. A sequence is an ordered list of numbers, and a series is the sum of terms of a sequence. Understanding the formulas of Arithmetic, Geometric, and Harmonic progressions helps in solving questions quickly and accurately.
1. Arithmetic Progression (A.P.)
An Arithmetic Progression is a sequence where the difference between any two consecutive terms is constant. This constant value is called the common difference (d).
(i) nth Term of A.P.
aₙ = a + (n – 1)d
Here,
a = first term
d = common difference
n = number of terms
(ii) Sum of n Terms
Sₙ = n/2 [2a + (n – 1)d]
(iii) If First and Last Term Known
Sₙ = n/2 (a + l)
where l = last termA.P. is widely used in exam questions involving number patterns, missing terms, and sum-based problems.
2. Geometric Progression (G.P.)
A Geometric Progression is a sequence where each term is obtained by multiplying the previous term by a constant called the common ratio (r).
(i) nth Term of G.P.
aₙ = a rⁿ⁻¹
(ii) Sum of n Terms (when r ≠ 1)
Sₙ = a (1 – rⁿ) / (1 – r)
(iii) Infinite G.P. (when |r| < 1)
S∞ = a / (1 – r)G.P. is commonly asked in questions involving growth, decay, multipliers, and ratio-based patterns.
3. Harmonic Progression (H.P.)
A sequence is said to be in Harmonic Progression if the reciprocals of its terms form an A.P.
nth Term of H.P.
If A.P. term = aₙ
then H.P. term = 1 / aₙH.P. is useful in questions of rates, averages, and inverse relationships.
4. Special Series Formulas
These formulas frequently appear in competitive exams:
(i) Sum of First n Natural Numbers
1 + 2 + 3 + … + n = n(n + 1) / 2
(ii) Sum of Squares
1² + 2² + … + n² = n(n + 1)(2n + 1) / 6
(iii) Sum of Cubes
1³ + 2³ + … + n³ = [n(n + 1) / 2]²This is known as the “square of sum of natural numbers” formula.
5. Averages: AM, GM, HM
These three means are very important for solving higher-level questions.
Arithmetic Mean (AM)
AM = (a + b) / 2
Geometric Mean (GM)
GM = √(ab)
Harmonic Mean (HM)
HM = 2ab / (a + b)Relationship:
AM ≥ GM ≥ HM (always true)
Why This Chapter Matters
Almost every exam—SSC, Banking, Railway, State PCS, Defence—asks questions directly from these formulas. Understanding the patterns and practicing both A.P. and G.P. problems helps in solving questions with speed and accuracy.