Prove that `loga+loga^(2)+loga^(3)+….+loga^(n)=(n(n+1))/2loga`

Prove log a + log a^(2) + "log" a^(3) +……..+ log a^(n) = (n (n + 1))/(2) log a

Show that log_a N + log (1)/(a)N=0

Prove that 1^(2)+2^(2)+3^(2)+.....+n^(2)=(n(n+1)(2n+1))/6

The value of loga b-log|b|= (a) loga (b) "log"|a| (c) -loga (d) none of these

Prove that ((n + 1)/(2))^(n) gt n!

By the principle of mathematical induction, prove that, for nge1 1^(3) + 2^(3) + 3^(3) + . . .+ n^(3)=((n(n+1))/(2))^(2)

For all n ge 1 prove that 1^(2)+2^(2)+ 3^(2)+4^(2)+….+n^(2)= (n(n+1)(2n+1))/(6)

Prove that (1 + i)^(n) + (1 - i)^(n) = 2^((n + 2)/(2)) cos (n pi)/(4)

If a_(1), a_(2), a_(3), ………, a_(n) ….. are in G.P., then the determinant Delta = |(""loga_(n)" "loga_(n + 1)" "log_(n + 2)""),(""loga_(n + 3)" "loga_(n + 4)" "log_(n + 5)""),(""loga_(n + 6)" "loga_(n + 7)" "log_(n + 8)"")| is equal to