AP And GP Sum n nth terms

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 =