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Arithmetic progression ( AP ) :

1)General term : t_n = a + (n-1)d

2)Sum of n terms :

S_n = n/2 ( 2a + (n-1)d )

3)Means :

(i)A.M of a and b = (a+b)/2

(ii) mth AM out of n A.M'sbetween a and b :

A_m= a + m[ (b-a)/(n+1) ]

(iii)A.M of a₁,a₂,a₃,..........a_n

A = a₁ + a₂ + ......... + a_n / n

Geometric means :

1)General term : t_n = ar^k-1

2)Sum to n terms :

S_n = a(rⁿ - 1)/r-1 , r 1

S_n = na , r = 1

S_infinity = a/(1-r) , lrl < 1

3)Means :

(i)GM of a and b = (ab)^1/2

(ii)m^th GM out on n GM's between a and b :

G_m = a(b/a)^m/n+1

(iii)GM of a₁,a₂,a₃.........a_n

= (a₁a₂a₃........a_n)^1/n

Harmonic progression (HP) :

1)General term : t_n = 1/a+(n-1)d with first term as 1/a

2)Means :

(i)H.M of a and b : H = 2ab/a+b

(ii)m^th HM out of n HM's between a and b :

H_m = ab(n+1)/m(a-b)+b(n+1)

(iii)H.M of a₁,a₂,a₃......a_n

H = [ n ] / [ 1/a₁ + 1/a₂ + 1/a₃ + ....... + 1/a_n ]

Arethmetico geometric progression (A.G.P) :

(i)General term : t_n = [a + (n-1)d]br^n-1

(ii)Sum to n terms :

S_n= [ab/1-r]+[dbr(1-r^n-1)/(1-r)²] + {[a+(n-1)d]brⁿ/1-r}

Where r 1

S_n = nb/2[2a + (n-1)d] where r = 1

S_infinity = [ab/1-r] + [dbr/(1-r)²] where lrl < 1

Relation between Means :

(i) For two numbers : G² = AH

Where G = GM ; H = HM ; A = AM

(ii) AM GM HM

Hope you all find it useful.

Cheers !!!!!!!!