`1/(n+1) + 1/(n+2)+…..+ 1/(3n+1) gt 1` for every positive integer n.

Show that P(m,1)+P (n,1)=P(M+n,1) for all positive integers m, n.

1/1.2 + 1/2.3 +…..+ 1/(n(n+1)) = n/(n+1)

Show that P(n,n)=P(n,n-1) "For all positive integers."

If a_1, a_2,........,a_n are in G.P. and a_i gt 0 for every i, then find the value of [[loga_n, loga_(n+1),loga_(n+2)],[loga_(n+1),loga_(n+2),loga_(n+3)],[loga_(n+2),loga_(n+3),loga_(n+4)]]

1 + 3 + 5 +……+ (2n -1) = n^2

Show that the function f: N rarr N, given by f(1) f(2)= 1 and f(x) = x - 1 for every x gt 2, is onto but not one-one.

7.5^(2n-1) + 2^(3n+1) is divisible by 17 for every natural n ge 1 .

4^(n+1) + 15n + 14 is divisible by 9 for every natural number n ge 0 .

3^(2(n-1)) + 7 is divisible by 8 for every natural n ge 2 .

Show that the curves y = 2^(x) and y = 5^(x) intersect at an angle tan^(-1) |("1n ((5)/(2)))/(1 + 1n 2 1n 5)| . Note Angle between two curves is the angle between their tangents at the point of intersection.