6. Find `log (a^n/b^n)+log (b^n/c^n) +log (c^n/a^n)` =?

If a, b, c, d are in G.P., prove that (a^n+b^n),(b^n+c^n),(c^n+a^n) are in G.P.

If a ,b ,c,d are in G.P. prove that (a^n+b^n),(b^n+c^n),(c^n+d^n) are in G.P.

If a ,b ,c,d are in G.P. prove that (a^n+b^n),(b^n+c^n),(c^n+d^n) are in G.P.

Assuming that log (mn) = log m + logn prove that log x^(n) = n log x, n in N

IfS_n=[1/(1+sqrt(n))+1/(2+sqrt(2n))++1/(n+sqrt(n^2))],t h e n(lim)_(n rarr oo)S_n is equal to (a)log 2 (b) log4 (c)log 8 (d) none of these

Given that lim_(nto oo) sum_(r=1)^(n) (log (r+n)-log n)/(n)=2(log 2-(1)/(2)) , lim_(n to oo) [(1)/(n^k)[(n+1)^k(n+2)^k.....(n+n)^k]]^(1//n) , is

Prove the following identities: (a) (log_(a) n)/(log_(ab) n) = 1+ log_(a) b" "(b) log_(ab) x = (log_(a) x log_(b) x)/(log_(a) x + log_(b) x) .

STATEMENT-1 : If a, b, c are distinct positive reals in G.P. then log_(a) n, log_(b) n, log_(c) n are in H.P. , AA n ne N and STATEMENT-2 : The sum of reciprocals of first n terms of the series 1 + (1)/(3) + (1)/(5) + (1)/(7) + (1)/(9) + ..... "is" n^(2) and

If a_1, a_2, a_3,....... are in G.P. then the value of determinant |(log(a_n), log(a_(n+1)), log(a_(n+2))), (log(a_(n+3)), log(a_(n+4)), log(a_(n+5))), (log(a_(n+6)), log(a_(n+7)), log(a_(n+8)))| equals (A) 0 (B) 1 (C) 2 (D) 3

If in an AP, t_1 = log_10 a, t_(n+1) = log_10 b and t_(2n+1) = log_10 c then a, b, c are in