Test the validity of the argument `(S_(1),S_(2),S)`, where `S_(1),p wedge q, S_(2), ~p and S:q.`

Let S_(1) be a square of side is 5 cm. Another square S_(2) is drawn by joining the mid-points of the sides of S_(1) . Square S_(3) is drawn by joining the mid-points of the sides S_(2) and so on. Then, (area of S_(1) + area of S_(2)+…+ area of S_(10) ) is a) 25(1-1/(2^(10)))cm^(2) b) 50(1-1/(2^(10)))cm^(2) c) 2(1-1/(2^(10)))cm^(2) d) 1-1/(2^(10))cm^(2)

The hydrogen-like species Ļi^(2+) is in a spherically symmetric state S_(1) with one radial node. Upon absorbing light the ion undergoes transition to a state S_(2) The state S_(2) has one radial node and its energy is equal to the ground state energy of the hydrogen atom. The state S_(1) is: 1s, 2s, 2p, 3s

The sets S_(1),S_(2),S_(3),… are given by S_(1)={(2)/(1)} , S_(2)={(3)/(2),(5)/(2)} , S_(3)={(4)/(3),(7)/(3),(10)/(3)} , S_(4)={(5)/(4),(9)/(4),(13)/(4),(17)/(4)},... Then, the sum of the numbers in the set S_(25) is

If S_(1), S_(2) , and S_(3) are, respectively, the sum of n, 2n and 3n terms of a G.P., then (S_(1)(S_(3)-S_(2)))/((S_(2)-S_(1)^(2)) is equal to

If S_(n) , denotes the sum of first n terms of a A.P. then (S_(3n)-S_(n-1))/(S_(2n)-S_(2n-1)) is always equal to