Mathematics: Sequence & Series Notes

ARJUNA JEE 2026 BATCH - MATHEMATICS SEQUENCE & SERIES

LECTURE OVERVIEW

• Lecture No.: 1

• Instructor: Sachin Jakhar (Sacchhu Sir)

TOPICS COVERED

1. Introduction

2. Arithmetic Progression (A.P.)

3. Question Practice

4. Rapid Fire Round

5. Properties of A.P., G.P., H.P., A.G.P.

INTRODUCTION TO SEQUENCES AND SERIES

• A sequence is a list of numbers in a specific order.

• A series is the sum of the terms of a sequence.

• Example sequences include: 1, 2, 3, 7, 8, 9, -11, 0, 5…

• General terms can be expressed in different forms, including arithmetic, geometric, and harmonic progressions.

ARITHMETIC PROGRESSION (A.P.)

• Definition: An A.P. is a sequence where the difference between consecutive terms is constant.

• Common Difference (d): Calculated as follows:

  • $d = T_{n+1} - T_n$

  • $T_n = a + (n-1)d$

  • Here, a is the first term, d is the common difference, and n is the term position.

• Examples of A.P.: 1, 3, 5; 7, 10, 13, etc.

• nth Term Formula:

  • The formula for the nth term of an A.P.:
    $T_n = a + (n-1)d$

SUM OF FIRST n TERMS OF A.P.

• The sum of the first n terms (denoted as S_n) is given by:
  $S_n = rac{n}{2} [2a + (n-1)d]$
  or
  $S_n = rac{n}{2} [T_1 + T_n]$

• Example:

  • For an A.P. with a = 2, d = 3, find the sum for the first 5 terms:

  • Terms are: 2, 5, 8, 11, 14

  • $S_5 = rac{5}{2} [2 + 14] = 5 imes 8 = 40$

PROPERTIES OF A.P.

1. If eight terms are in A.P., then:

   • $2b = a + c$

2. The sum of n terms (n even) is equal to twice the sum of the first half terms.

3. Reciprocal properties: If a, b, c are in A.P., then:

   • Reciprocals: $rac{1}{a}, rac{1}{b}, rac{1}{c}$

   • Are not necessarily in A.P.

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 (r).

• Example: 2, 4, 8, 16 (where r = 2)

• General term of a G.P.:
  $T_n = ar^{n-1}$

SUM OF INFITE TERMS OF A G.P.

• Given a geometric series where $|r| < 1$, the formula for the sum is:
  $S_{ ext{infinite}} = rac{a}{1 - r}$

HARMONIC PROGRESSION (H.P.)

• A sequence where the reciprocals of the terms form an A.P.:

  • Example: a, b, c are in H.P. if
    $rac{1}{a}, rac{1}{b}, rac{1}{c} ext{ are in A.P.}$

ARITHMETIC GEOMETRIC PROGRESSION (A.G.P.)

• Combination of A.P. and G.P.

• Structure: a, (a + d), ar, (ar + d), … which combines terms from both progressions.

QUESTION PRACTICE

1. Fast question set on A.P., G.P., and H.P.

2. Practice basic and advanced problems based on definitions and properties.

RAPID FIRE ROUND

• Quick questions to test comprehension and speed regarding sequences and series.

HOMEWORK

• Review properties of sequences and series; solve provided exercises from last class.

CONCLUSION

• Emphasize continuous practice to understand concepts well.

• Reminder of the importance of learning strategies for tackling series-related questions in exams.