Arithmetic progression(A.P.), Geometric progression (G.P), Harmonic progression (H.P) , nth term , sum of n terms, of A.P. and G.P sum of infinite number of terms of a G.P , sum of the first n natural numbers, sum of the squares of the first n natural numbers and sum of the cubes of the first n natural numbers



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Progressions



Arithmetic Progressions (A. P.)

The sequence a , (a +d ), (a +2d ), (a +3d ), (a +4d ), . . . is called an arithmetic progression.

' a ' is the first term and ' d ' is the common difference of the A.P.

The nth term of the A.P. is a + ( n-1)d

The sum of n terms of the A.P. is = [ 2a + (n - 1 ) d ]

or = [ + ] where and are the first and nth terms of the A.P.

Three numbers a , b, c are in A.P. if 2b = a +c.



Geometric Progression (G.P. )

The sequence a , a r, , . . . is called a geometric progression.

' a' is the first term and ' r ' is the common ratio of the geometric progression.

The nth term of the G.P. is =

The sum of n terms of the G.P. is =

or = .

Three numbers a , b, c are in G.P. if .

The sum of infinite number of terms of the G.P is = , provided .



Harmonic Progression (H.P.)

The sequence , . . . are said to be in Harmonic Progression if their reciprocals ,. . . are in Arithmetic Progression.



Sum of the first n natural numbers, 1+2+3+ ... + n =



Sum of squares of the first n natural numbers, 12+22+32+ ... + n2 =



Sum of the cubes of the first n natural numbers, 13+23+33+ ... + n3 =













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