Study Notes on Sequence and Series

Instructor Background

• Sachin Mor Sir
  • Senior Mathematics Faculty
  • 10+ Years of JEE Teaching Experience
  • Mentored over 10,000 students for JEE Advanced
  • Helped numerous students secure Top 100 ranks in JEE

Overview of JEE 2026 Mathematics

• Focus on Sequences and Series
• Content structured in a detailed one-shot approach covering key concepts

Key Concepts Covered

• Types of Progressions:

  • Arithmetic Progression (A.P.)
  • Geometric Progression (G.P.)
  • Harmonic Progression (H.P.)
  • Arithmetic-Geometric Progression (A.G.P.)

• Means:

  • Arithmetic Mean (A.M.)
  • Geometric Mean (G.M.)
  • Harmonic Mean (H.M.)

Summation Formulas

• Sum of first n natural numbers:
  $ext{Sum} = \frac{n(n+1)}{2}$
• Sum of squares of the first n natural numbers:
  $ext{Sum} = \frac{n(n+1)(2n+1)}{6}$
• Sum of cubes of the first n natural numbers:
  $ext{Sum} = \frac{n^2(n+1)^2}{4}$

Sequence and Series Definitions

• Sequence: An ordered list of numbers.
• Series: The summation of the terms of a sequence.
• Notation:
  • $T_n$ denotes the n-th term of any sequence.
  • $S_n$ denotes the summation of the first n terms of any series.

Arithmetic Progression (A.P.)

• Definition: A sequence where the difference between consecutive terms remains constant.

• General form:
  $a, a + d, a + 2d, ext{…}, a + (n - 1)d$

  • Where a is the first term, and d is the common difference.

• n-th Term Formula:
  $T_n = a + (n-1)d$

• Sum of first n terms (S_n):
  $S_n = \frac{n}{2} [2a + (n - 1)d]$

  • Alternatively,
    $S_n = \frac{n}{2} [T_1 + T_n]$

Properties of A.P.

• If two sequences are in A.P., then their sums/subtractions are also in A.P.
• If each term of an A.P. is increased or decreased by the same number, the resulting sequence retains the A.P. structure with the same common difference.
• If each term is multiplied or divided by a non-zero number { t k}:
  • New common difference:
  • Multiplied by k: $kd$
  • Divided by k: $\frac{d}{k}$
• k-th term from the last:
  $T_{k} ext{ from last} = T_{(n-k+1)} ext{ from beginning}$

Special Arithmetic Progressions

• Properties of specific sets of numbers in A.P.:

  • Example 1: Taking three numbers in A.P.:
    $a - d, a, a + d$
  • Example 2: Five numbers in A.P.:
    $a - 2d, a - d, a, a + d, a + 2d$
  • Example 3: Four numbers in A.P.:
    $a - 3d, a - d, a + d, a + 3d$

• Picking terms from A.P. in an interval leads to another A.P.

Problem Solving Examples

• Example 1: For the A.P. terms: $a_1, a_2, a_3, ext{…}, a_n$ given a4 - a7 + a10 = m, find the sum of the first 13 terms of this A.P.
  • Options provided:
    • (A) 15m
    • (B) 10m
    • (C) 12m
    • (D) 13m

Geometric Progression (G.P.)

• A sequence where each term after the first is found by multiplying the previous term by a fixed non-zero number called the common ratio.
  • General form:
    $a, ar, ar^2, ext{…}, ar^{(n-1)}$
  • G.P. Properties:
  • If each term of a G.P. is raised to a power k, the resulting series is also a G.P.
• Sum of first n terms (S_n):
  For |r| < 1:
  $S_n = \frac{a(1 - r^n)}{1 - r}$

Harmonic Progression (H.P.)

• A sequence is in H.P. if the reciprocals of its terms are in A.P.
• H.P. Properties:
  • If $a_1, a_2, a_3, ext{…}$ is in H.P., then
    $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, ext{…}$ must be in A.P.

Means

• Arithmetic Mean (A.M.):
  Between two numbers $a$ and $b$, the A.M. is given by:
  $A.M. = \frac{a + b}{2}$
• Geometric Mean (G.M.):
  For two positive numbers $a$ and $b$,
  $G.M. = ext{sqrt}(ab)$
• Harmonic Mean (H.M.):
  For numbers $a$ and $b$,
  $H.M. = \frac{2ab}{a + b}$

Conclusion

• This study guide captures essential formulas, definitions, example problems, and properties of sequences and series relevant for JEE Mathematics. It aims to provide a comprehensive understanding and facilitate effective study preparations.