Sequence And Series Class 11 Notes Mathematics Chapter 9 - CBSE

Chapter : 9

What Are Sequence And Series ?

Sequence

By a sequence, we mean an arrangement of number in definite order according to some rule. Also we define a sequence as a function whose domain is the set of natural numbers or some subsets of the type {1, 2, 3......k}. A sequence containing, a finite number of terms is called a finite sequence. A sequence is called infinite if it is not a finite sequence.

Series

Let a₁ , a₂ , a₃ ..... be the sequence, then the sum expressed as a₁ + a₂ + a₃ + .... is called a series. A series is called finite series if it has delete finite number of terms.

Arithmetic Progression (A.P.)

A sequence a₁ , a₂ , a₃...... a_n is called arithmetic progression if a_{n + 1} = a_n + d , n ∈ N where a1 is called the first term and the constant term d is called the common difference of the A.P. The general term or n^th term of an A.P. is given by a_n = a + (n – 1)d.

The sum of first n terms of an A.P. is given by

$$\text{S}_{n} =\frac{n}{2}[2a + (n-1)d]\\=\frac{n}{2}[a + l]$$

where l = last term

Arithmetic Mean

The arithmetic mean of any two numbers a and b is given by $$\frac{a + b}{2}\space\text{i.e., the sequence a,}$$ A.M., b is in A.P.

Geometric Progression

A sequence is said to be a Geometric Progression or G.P. if the ratio of any term to its preceding term is same throughout. This common factor is called common ratio denoted by r. We denoted the first term of G.P. by a and its common ratio by r. The general or nth term of G.P. is given by

a_n = ar^{n – 1}

The sum S_n of first n terms of G.P. is given by

$$\text{S}_{n} =\frac{a(r^{n}-1)}{r-1}\text{or}\\\frac{a(1 - r^{n})}{1-r}\space\text{if r}\neq 1\\\text{if}(r\gt 1)\text{if}\space(\therefore r\lt 1)$$

Geometric Mean

The geometric mean (G.M.) of any two positive numbers a and b is given by

$$\text{G.M. =}\sqrt{\text{ab}}$$

The sequence a; G.M. b is G.P.

Sum Of n Terms Of Special Series

1 + 2 + 3 +.........+ n [Sum of first n natural numbers]

S_n = 1 + 2 + 3 + ...... n

$$\text{S}_{n}=\frac{n(n+1)}{2}$$

1² + 2² + 3² + ...... + n² [Sum of squares of the first n natural numbers]

$$\text{Then\space} S_{n} =\frac{n(n+1)(2n+1)}{6}$$

1³ + 2³ + 3³ + ...... + n³ [Sum of cubes of first n natural numbers]

$$\text{S}_{n}=\begin{bmatrix}\frac{n(n+1)}{2}\end{bmatrix}^{2}$$

Relationship Between A.m. And G.m.

Let A and G be A.M. and G.M. of two given positive real numbers a and b, respectively. Then

$$\text{A} =\frac{a+b}{2}\space\text{and G =}\sqrt{\text{ab}}\\\text{Thus, we have}\\\text{A - G}=\frac{a+b}{2}-\sqrt{ab}\\=\frac{a+b- 2\sqrt{ab}}{2}\\=\frac{(\sqrt{a}-\sqrt{b})^{2}}{2}\geq 0\\\text{Thus,A}\geq G.$$