For a positive integer n let `a(n) = 1 + (1)/(2) + (1)/(3) + (1)/(4) + ...+ (1)/((2^(n)) -1)`, then

For a positive integer n let a(n)=1+1/2+1/3+1/4+1/((2^n)-1)dot Then a(100)lt=100 b. a(100)dot> 100 c. a(200)lt=100 d. a(200)lt=100

For and odd integer n ge 1, n^(3) - (n - 1)^(3) + …… + (- 1)^(n-1) 1^(3)

Find the least positive integer n such that ((1 + i)/(1 - i))^(n) = 1 .

For a fixed positive integer n , if =|n !(n+1)!(n+2)!(n+1)!(n+2)!(n+3)!(n+2)!(n+3)!(n+4)!| , then show that [/((n !)^3)-4] is divisible by ndot

Find the least positive integer n such that ((1+i)/(1-i))^(n)=1 .

Let a_(n) be the n^(th) term of a G.P of positive integers. Let sum_(n = 1)^(100) a _(2n) = alpha and sum_(n = 1)^(100) a_(2n +1) = beta such that alpha != beta . Then the common ratio is

If n_1, n_2 are positive integers, then (1 + i)^(n_1) + ( 1 + i^3)^(n_1) + (1 + i_5)^(n_2) + (1 + i^7)^(n_2) is real if and only if :

For any positive integer (m,n) (with ngeqm ), Let ((n),(m)) =.^nC_m Prove that ((n),(m)) + 2((n-1),(m))+3((n-2),(m))+....+(n-m+1)((m),(m))

If n is a positive integer, prove that 1-2n+(2n(2n-1))/(2!)-(2n(2n-1)(2n-2))/(3!)++(-1)^(n-1)(2n(2n-1)(n+2))/((n-1)!)= (-1)^(n+1)(2n)!//2(n !)^2dot