Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 10 Sequences and Series will enable students to study smartly.

## CHSE Odisha 11th Class Math Notes Chapter 10 Sequences and Series

**Sequence:**

A sequence is a function whose domain is N (The set of natural numbers).

Note: We can use the set of whole numbers as a domain.

Real Sequence:

If the range of a sequence is a subset of ‘R’, then it is a real sequence.

⇒ If: N → R is a sequence then f(n) for n = 1, 2, 3, ….. are the terms of the sequence.

Finite and infinite sequence:

A sequence with a finite number of terms is a finite sequence otherwise it is infinite.

Note: We denote a sequence by (t_{n}) or {t_{n}} where f(n) = t_{n}

**Series:**

An expression of the type t_{1} + t_{2} + t_{3} + ….. (or ∑t_{n}) where t_{n} is the nth term of a sequence is a series.

Partial sums:

If \(\sum_{n=1}^{\infty} t_n\) is a series then a sum \(\mathrm{S}_n=\sum_{k=1}^n t_k\) is called the nth partial sum of the series for n = 1, 2, 3 …..

∴ s_{1} = t_{1}, s_{2} = t_{1} + t_{2}, s_{3} = t_{1} + t_{2} + t_{3} and so on.

**Progression:**

Progression is a sequence whose terms follow as pattern.

Arithmetic progression (A.P):

A sequence (t_{n}) is an A.P. If t_{n+1} – t_{n} = d (constant) for n = 1, 2, 3, …..

(a) General form: a, a + d, a + 2d, a + 3d …..

(b) n^{th} term: t_{n} = a + (n – 1)d, where t_{1} = a, and the common difference = d

(c) Sum of first n terms (Sn):

S_{n} = \(\frac{n}{2}\)[2a + (n – 1)d]

= \(\frac{n}{2}\)[a + l]

where a = first term

d = common difference

l = last term (or nth term)

Note:

1. If a, b, c are in A.P. then 2b = a + c.

2. If 3 numbers are in A.P. then we take them as a – d, a, a + d.

3. If 4 numbers are in A.P. then we take rhem as a – 3d, a – d, a + d, a + 3d.

(d) Insertion of arithmetic means between two given number:

Let m_{1}, m_{2}, m_{3} …. m_{n} are ‘n’ arithmetic means between ‘a’ and ‘b’ then m_{k} = a + \(\frac{k(b-a)}{n+1}\) for k = 1, 2, ….. n.

Geometric progression (G.P):

If \(\frac{t_{n+1}}{t_n}\) = r (constant), for n = 1, 2, 3, ….. then the sequence (t_{n}) is a geometrical progression.

(a) General form: a, ar, ar^{2}, ar^{3} …..

(b) nth term of GP: nth term of G.P. = t_{n} = ar^{n-1}.

(c) sum of first n terms of a G.P.: S_{n} = \(\frac{a\left(1-r^n\right)}{1-r}(\text { for } r \neq 1)\)

(d) sum of an infinite G.P.: If |r| < 1 then the sum of the infinite G.P. a, ar, ar^{2} ….. is S_{∞} = \(\frac{a}{1-r}\)

Note:

1. If a, b, c are in G.P. then b^{2} = ac

2. If 3 numbers are in G.P. we take them as \(\frac{a}{r}\), a, ar.

3. If 4 numbers are in G.P. then we take them as \(\frac{a}{r^3}, \frac{a}{r}, a r, a r^3\)

(e) Insertion of geometric means between two numbers:

If g_{1}, g_{2} ……g_{n} are n geometric means between a and b then g_{k} = a \(\left(\frac{b}{a}\right)^{\frac{k}{n+1}}\), k = 1, 2, 3, …. n

**Harmonic Progression (H.P):**

A sequence a_{1}, a_{2}, a_{3} ….. of non zero numbers is called a Harmonic progression if the sequence \(\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}\) ….. is an A.P.

(a) Harmonic mean:

Harmonic mean(H) between two numbers a and b is \(\frac{1}{\mathrm{H}}=\frac{\left(\frac{1}{a}+\frac{1}{b}\right)}{2}\)

= \(\frac{a+b}{2 a b}\)

⇒ H = \(\frac{2 a b}{a+b}\)

(b) Insertion of n harmonic means between two numbers:

Let H_{1}, H_{2} ….. H_{n} are n harmonic means between a and b then \(\frac{1}{\mathrm{H}_K}=\frac{1}{a}\) + k_{D}, where D = \(\frac{a-b}{(n+1) a b}\).

Relation among A.M., G.M. and H.M.

AM ≥ GM ≥ HM

Arithmetic co-geometric sequence(AGP):

If (a_{n}) is an A.P. and (b_{n}) is an G.P. then the series (a_{n}b_{n}) is called an arithmetic co-geometric sequence.

(a) General form: a, (a + d) r, (a + 2d) r^{2}, (a + 3d)r^{3},…..

(b) nth term of A.G.P.: t_{n} = {a + (n – 1)d} r^{n-1}.

(c) sum of first terms of A.G.P.:

The sum of first n terms of A.G.P. a, (a + d) r, (a + 2d) r^{2}, ….. is

S_{n} = \(\frac{a}{1-r}+d r\left(\frac{1-r^{n-1}}{(1-r)^2}\right)-\frac{[a+(n-1) d] r^n}{1-r}\) for r ≠ 1

(d) sum of infinite A.G.P.: If |r| < 1 then we have \(S_{\infty}=\frac{a}{1-r}+\frac{d r}{(1-r)^2}\)

Sum of special sequences.:

**Binomial Series:**

(a) Binomial theorem for any real index:

- (1 + x)
^{n}= 1 + nx + \(\frac{n(n-1)}{2} x^2+\frac{n(n-1)(n-2)}{3 !} x^3+\ldots\) for |x| < 1 - (1 – x)
^{-1}= 1 + x + x^{2}….. - (1 + x)
^{-1}= 1 – x + x^{2}– x^{3}….. - (1 + x)
^{-2}= 1 – 2x + 3x^{2}– 4x^{3}….. - (1 – x)
^{-2}= 1 + 2x + 3x^{2}+ …..

Exponential series:

Logarithmic Series: