In general, a sequence is an ordered list of numbers or items. 

In mathematical terms, 'sequence' is defined as a function that has a domain belonging to a set of natural numbers. 

The numbers or items of sequence are called 'elements' or 'terms'.

When we add all the elements or terms of the sequence, we get a series. The value we obtain by adding all the elements or terms of a sequence is called the value of the series.  For example, the sequence "2, 3, 4,5" contains the terms "2", "3", "4", and "5"; the corresponding series is the sum " 2 + 3 + 4+5", and the series' value is 14.

An arithmetic progression is a sequence of numbers. In A.P, each successive term is equal to the sum of all of its preceding terms and a fixed number, called the common difference(d). 

The progression of A.P is decided by the fixed number. If the fixed number is a positive value, then A.P will increase; if it’s a negative integer, A.P will decrease. 

The first term of A.P is represented by  ‘a’.

The N^th term of an AP:

• • t_n= a+(n−1)d

Where, d = a_n−a_n−1

Sum of first N terms of an AP:

S_n= n/2 [a+(n−1)d]= n/2[a+l]

Where,l is last term of an A.P

G.P is also a type of sequence. In G.P, every term is calculated by multiplying its preceding term by a fixed constant number called the common ratio. 

First-term of GP is always a nonzero integer.

Common ratio = term/its preceding term  

a = first term 

r= common ratio 

Then, 

G.P = a,ar,ar²,ar³,ar⁴,.......

The Nth term of a GP:

t_n =ar_n−1

Sum of first N terms of a GP: S_n=a(1−rn)/(1−r),r≠1

Sum of infinite GP when |r|<1&n→∞

|r|<1⇒rn→0⇒S_°°=a/1−r

For two positive numbers, let say m and n, the arithmetic mean is always equal to or greater than the geometric mean, and geometric mean is equal to  √mn.

AM or Arithmetic Mean is the average of the sets of numbers of the data. It is calculated by adding all the elements of the data divided by the total number of elements in the data.

GM or Geometric Mean is the mean value or the central term in the set of numbers in geometric progression.

 

There are a total of three special series. 

1. 1. 1 + 2 + 3 +… + n (sum of first n natural numbers)

   2. 12 + 22 + 32 +… + n2(sum of squares of the first n natural numbers)

   3. 13 + 23 + 33 +… + n3(sum of cubes of the first n natural numbers)

Sum to n terms of first n natural numbers: 

 

Sum of First n Natural Numbers (1 + 2 + 3 +… + n)

This is an A.P. 

Here,  a = 1, d =1 

Formula of the special series

Sn = n/2 [2a + ( n-1 )d]

Sn = n/2 [2× 1 + ( n-1 )1]

= n/2 [ 2+n-1]

= n/2 [n+1]

 

Here are some trigonometry Ncert solutions class 11

Q1. tan x = √3

Solution:

Q2 . sin 2x + cos x = 0

Solution:

It is given that

sin 2x + cos x = 0

We can write it as

2 sin x cos x + cos x = 0

cos x (2 sin x + 1) = 0

cos x = 0 or 2 sin x + 1 = 0

Let cos x = 0

 

     1. What are the basic trigonometric functions?

 The six basic trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).

     2. How many exercises are there in Chapter 3 of NCERT Solutions for Class 11 Maths?

Altogether, there are four exercises in chapter 3 of NCERT Class 11 maths, including one miscellaneous exercise.

     3. How are Radians and Degrees related?

Degree measures angle in 2 dimensions or upto 360°.

 Radians measure angles in 3 dimensions. It is represented by (theta). 

Radian = arc length (s) / radius(r). 

The relation between Radian and degree is given in terms of a circle.

360° =2pi radian(theta).