For as Ap. `S_(n)= m and S_(m) = n` . Prove that `S_(m+n)=-(m+n).(m ne n)`

For an A.P., S_(1)=6 and S_(7)= 105 . Prove that S_(n): S_(n-3) = (n+ 3): (n-3)

S_(n) is the sum of n terms of an A.P. If S_(2n)= 3S_(n) then prove that (S_(3n))/(S_(n))= 6

If in an AP, S_(n)= q n^(2) and S_(m)= qm^(2) , where S_(r ) denotes the sum of r terms of the AP, then S_(q) equals to,

For an AP, if a_(n)=4, d=2 and S_(n)= -14 , find n and a.

If S_(1), S_(2) and S_(3) be respectively the sum of n, 2n and 3n terms of a G.P. Prove that S_(1) (S_(3)-S_(2))= (S_(2)-S_(1))^(2) .

For an A.P., show that t_(m) + t_(2n + m) = 2t_( m + n)

If x^(m).y^(n)= (x + y)^(m+n) , prove that (dy)/(dx)= (y)/(x)

If n(A) = m and n(B) = n then numbers of function f: A rarr B are m.n .

Let the sum of n, 2n, 3n terms of an A.P. be S_1, S_2 and S_3 , respectively, show that S_3 =3(S_2-S_1)

In an AP: (i) Given a=5, d=3, a_(n) = 50 , find n and S_(n) (ii) given a=7, a_(13) = 35 , find d and S_(13) . (iii) given a_(12) = 37, d=3 , find a and S_(12) (iv) given a_(3) = 15, S_(10) = 125 , find d and a_(10) (v) given a=2, d= 8, S_(n) = 90 , find n and a_(n) (vi) given a_(n) = 4, d=2, S_(n) = - 14 , find n and a. (vii) given l=28, S= 144, and there are total 9 terms, find a.