Show that the linear function in n i.e. f(n)=an+b determines an arithmetic progression, where a,b are constants.

The nth term of a series is given by t_(n)=(n^(5)+n^(3))/(n^(4)+n^(2)+1) and if sum of its n terms can be expressed as S_(n)=a_(n)^(2)+a+(1)/(b_(n)^(2)+b) where a_(n) and b_(n) are the nth terms of some arithmetic progressions and a, b are some constants, prove that (b_(n))/(a_(n)) is a costant.

Show that the function f: RrarrR given by f(x)= ax + b where a, b in R, ane0 is a bijection.

Show that the function f:N rarr N given by f(x) = 5x is one-one but not onto.

Show that the function f:N rarr N given by f(x) = 4x is one-one, but not onto.

Show that the function f: N rarrN given by f(x)= 3x is one-one but not onto.

Show that function f: Nrarr N given by f(x) = x^2 is neither one-one nor onto.

ABC is a triangle such that : sin (2A +B) = sin (C-A) = -sin (B+ 2C) = 1/2 . If A, B and C are in arithmetical progression, determine the values of A, B and C.

Integrate the function: f'(ax + b) [f (ax + b)]^n

Find out which of the following sequences are arithmetic progressions. For those which are arithmetic progressions, find out the common difference. :- a + b, (a + 1) + b, (a + 1) + (b + 1), (a + 2) + (b + 1), (a + 2) + (b + 2)…….

If the coefficients of a^(r-1), a^(r ), a^(r+1) in the binomial expansion of (1 + a)^(n) are in Arithmetic Progression, prove that: n^(2) -n(4r+1) + 4r^(2) -2=0