Sequences and Series

Master arithmetic and geometric progressions, means, and series summation for JEE Mathematics.

Sequences and series deal with ordered lists of numbers and their sums.

Overview

graph TD
    A[Sequences] --> B[AP]
    A --> C[GP]
    A --> D[Special Series]
    B --> B1[nth term]
    B --> B2[Sum]
    C --> C1[nth term]
    C --> C2[Sum]

Arithmetic Progression (AP)

A sequence where consecutive terms have constant difference.

$$a, a+d, a+2d, ..., a+(n-1)d$$

nth Term

$$\boxed{a_n = a + (n-1)d}$$

Sum of n Terms

$$\boxed{S_n = \frac{n}{2}[2a + (n-1)d] = \frac{n}{2}(a + l)}$$

where l = last term

Properties

  1. If a, b, c are in AP: $2b = a + c$
  2. AM of a and b: $A = \frac{a+b}{2}$
  3. Three terms in AP: $a-d, a, a+d$
  4. Four terms: $a-3d, a-d, a+d, a+3d$

Geometric Progression (GP)

A sequence where consecutive terms have constant ratio.

$$a, ar, ar^2, ..., ar^{n-1}$$

nth Term

$$\boxed{a_n = ar^{n-1}}$$

Sum of n Terms

$$\boxed{S_n = a\frac{r^n - 1}{r - 1} = a\frac{1 - r^n}{1 - r}}$$

(r ≠ 1)

Sum to Infinity (|r| < 1)

$$\boxed{S_\infty = \frac{a}{1-r}}$$

Properties

  1. If a, b, c are in GP: $b^2 = ac$
  2. GM of a and b: $G = \sqrt{ab}$
  3. Three terms: $\frac{a}{r}, a, ar$
JEE Tip
For product of terms in GP, use symmetric form: a/r, a, ar or a/r², a/r, a, ar, ar²

Harmonic Progression (HP)

Reciprocals form an AP.

$$\frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2d}, ...$$

nth term of HP: Reciprocal of nth term of corresponding AP

HM of a and b: $H = \frac{2ab}{a+b}$

Relationship Between Means

For two positive numbers a and b:

$$A \geq G \geq H$$ $$G^2 = AH$$

Arithmetico-Geometric Progression (AGP)

Product of AP and GP terms:

$$a, (a+d)r, (a+2d)r^2, ...$$

Sum: Multiply by r, subtract, and solve.

Special Series

Sum of First n Natural Numbers

$$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$$

Sum of Squares

$$\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$$

Sum of Cubes

$$\sum_{k=1}^{n} k^3 = \left[\frac{n(n+1)}{2}\right]^2$$

Method of Differences

If $a_n = f(n) - f(n-1)$:

$$S_n = f(n) - f(0)$$

Telescoping series

Practice Problems

  1. Find the sum of first 20 terms of AP: 1, 4, 7, 10, …

  2. Find the sum to infinity of GP: 1, 1/2, 1/4, 1/8, …

  3. If AM of two numbers is 5 and GM is 4, find the numbers.

  4. Find sum: $1·2 + 2·3 + 3·4 + ... + n(n+1)$

Quick Check
Why is $A ≥ G$ for positive numbers?

Further Reading