`tan (A_1+A_2 +... + A_n) = (S_1-S_3+S_5-S_7+...) / (1 - S_2 + S_4 - S_6+...)`

If a_1. a_2 ....... a_n are positive and (n - 1) s = a_1 + a_2 +.....+a_n then prove that (a_1 + a_2 +....+a_n)^n ge (n^2 - n)^n (s - a_1) (s - a_2)........(s - a_n)

Tan (A1+A2+A3+a4)=(S1-S3+S5-S7+)/(1-S2+S4-S6+..):: Refer-Cengage -3.12:: Refer-Trigonometric Ratios Of Multiple And Sub-Multiple Angles

If S_1 , S_2 and S_3 are respectively the sum of n, 2n and 3n terms of a G.P., then prove that S_1(S_3 - S_2) = (S_2 - S_1)^2 .

If S_(1), S_(2), S_(3),….., S_(n) are the sum of infinite geometric series whose first terms are 1,3,5…., (2n-1) and whose common rations are 2/3, 2/5,…., (2)/(2n +1) respectively, then {(1)/(S_(1) S_(2)S_(3))+ (1)/(S_(2) S_(3) S_(4))+ (1)/(S_(3) S_(4)S_(5))+ ........."upon infinite terms"}=

If S_(1), S_(2), S_(3),….., S_(n) are the sum of infinite geometric series whose first terms are 1,3,5…., (2n-1) and whose common rations are 2/3, 2/5,…., (2)/(2n +1) respectively, then {(1)/(S_(1) S_(2)S_(3))+ (1)/(S_(2) S_(3) S_(4))+ (1)/(S_(3) S_(4)S_(5))+ ........."upon infinite terms"}=

Consider the sequence of natural numbers S_0,S_1,S_2 ,... such that S_0 =3, S_1 = 3 and S_n = 3 + S_(n-1) S_(n-2) , then

If S_(1), S_(2), S_(3) be respectively the sums of n, 2n and 3n terms of a G.P., prove that, S_(1)(S_(3) - S_(2)) = (S_(2) - S_(1))^(2) .

Let S_n=1+2+3++n and P_n=(S_2)/(S_2-1)*(S_3)/(S_3-1)*(S_4)/(S_4-1)* . . . *(S_n)/(S_n-1) Where n in N ,(ngeq2)dot Then lim_(n→oo)P_n=______