An arithmetic progression (A.P.) or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. For instance, the sequence 3, 5, 7, 9, 11, 13… is an arithmetic progression with common difference 2.

If the initial term of an arithmetic progression is a₁ and the common difference of successive members is d, then the nth term of the sequence is given by:

and in general

A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression.

Question:-

Find the 25th term of the following arithmetic progression
3, 6, 9, 12, 15, …

Answer:-

3, 6, 9, 12, 15, …

a = 3 , d = 3 , n = 25

Tn = a + (n − 1)d

T25 = 3 + (25 − 1)(3)

= 3 + 72

= 75

∴ the 25th term of the A.P. is 75.

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Topic:-Trigonometry

Trigonometry is a branch of mathematics that deals with triangles, particularly those plane triangles in which one angle has 90 degrees (right triangles). Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships.

This trigonometry help show us how to solve a problem using Tan formulas.

Question:-

 Solve 

  Cos2x=cosx

Answer:-

Lets write 

cos2x as 2 cos2x-1

So we have 

   2 cos2x-1=cosx

  Then subtract cosx on both sides

2cos2x-1-cosx=cosx-cosx

2cos2x-1-cosx= 0

Lets take cosx as y

So the equaton become

  2y2-y-1= 0

  (y-1)(2y+1)=0

  y-1=0 or 2y+1=0
                 -1
  So y = 1 or y= ------
                  2

Since y=cosx
                    -1
 Cosx=1   or  cosx= ----
                     2
We know that cos0=0 

So cosx=cos0

 x=0 ,

Similarly for the second equation 

 x=120 or 240

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