If `T_n` denotes the nth term of a G.P., prove that `(T_n)^2 = T_(n+r) T_(n-r)`

If m times the mth term is equal to n times the nth term of an A.P. prove that(m+n)th term of an A.P. is zero.

If S_n denotes the sum of n terms of a G.P. whose first term and common ratio are a and r respectively, then : S_1 +S_3+S_5+.....+S_(2n-1)= (an)/(1-r)-(ar(1-r^(2n)))/((1-r)^2 (1+r)) .

Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P^(2)R^(n) = S^(n) .

If sum_(r=1)^(n)T_(r)=(n(n+1)(n+2)(n+3))/(12) where T_(r) denotes the rth term of the series. Find lim_(nto oo) sum_(r=1)^(n)(1)/(T_(r)) .

Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P^2R^n = S^n .

If S_n represents the sum of n terms of a G.P. whose first term and common ratio are a and r respectively. Prove that : S_1 +S_2+S_3+.....+S_(m)= (am)/(1-r)-(ar(1-r^(m)))/((1-r)^2) .

If t_r is the rth term is the expansion of (1 + a)^n , in ascending powers of a, prove that : r(r+1) t_(r+2)= (n-r+1)(n-r) a^2 t_r .

If the first and the nth term of a G.P. are a and b. respectively, and if P is the product of n terms, prove that P^2 = (ab)^n .

Let M and N be two 3xx3 nonsingular skew-symmetric matrices such that Mn=NM . If P^(T) denotes the transpose of P, then M^(2) N^(2) (M^(T)N)^(-1) (MN^(-1))^(T) is equal to

The numbers 1,3,6,10,15,21,28"..." are called triangular numbers. Let t_(n) denote the n^(th) triangular number such that t_(n)=t_(n-1)+n,AA n ge 2 . The number of positive integers lying between t_(100) and t_(101) are: