Show that `((1)/(r )-(1)/(r_(1)))((1)/(r_(1))-(1)/(r_(2)))((1)/(r_(2))- (1)/(r_(3)))=(1 6 R)/(r ^(2)(suma)^(2))`

Show that (b-c)/(r _(1))+ (c-a)/(r _(2))+(a-b)/(r _(3)) =0.

((n),(r))+((n),(r-1))=((n+1),(r-1))

Prove that ((n),(r))+2((n),(r-1))+((n),(r-2))=((n+2),(r))

((n),(r))+2.((n),(r-1))+((n),(r-2))=((n+2),(r))

Two resistors of resistances R_(1)=100pm3 ohm and R_(2)=200pm4 ohm are connected (a) series , (b) in parallel. Find the equivalent resistance of the (a) series combination, (b) parallel combination. Use for (a) the relation R = R_(1)+R_(2) and for (b) (1)/(R') = (1)/(R_(1))+(1)/(R_(2)) and (DeltaR')/(R'^(2)) = (DeltaR_(1))/(R_(1)^(2))+(DeltaR_(2))/(R_(2))

The Bohr model for the H-atom relies on the Coulomb's law of electrostatics. Coulomb's law has not directly been verified for very short distances of the order of angstroms. Supposing Coulomb's law between two opposite charge +q_(1),-q_(2) is modified to |F|=(q_(2)q_(1))/((4pi epsi_(0)))(1)/(r^(2)),r ge R_(0) =(q_(1)q_(2))/(4pi epsi_(0))(1)/(R_(0)^(2))((R_(0))/(r))^(e),r lt R_(0) Calculate in such a case, the ground state energy of a H-atom if epsi=0.1 R_(0)=1Å

Let R_(t) represents activity of a sample at an insant and N_(t) represent number of active nuclei in the sample at the instant. T_(1//2) represents the half life. {:(,"Column I",,"Column II"),((A),t=T_(1//2),(p),R_(t)=(R_(0))/(2)),((B),t=(T_(1//2))/(ln2),(q),N_(0)-N_(t)=(N_(0))/(2)),((C),t=(3)/(2)T_(1//2),(r),(R_(t)-R_(0))/(R_(0)) = (1-e)/(e)),(,,(s),N_(t)=(N_(0))/(2sqrt(2))):}

Show that |{:(sum_(r=1)^(16)2^r,a,2(2^(16)-1)),(3sum_(r=1)^(16)4^r,b,4(4^(16)-1)),(7sum_(r=1)^(16)8^r,c,8(8^(16)-1)):}|=0

The distance between two particles of masses m_(1)andm_(2) is r. If the distance of these particles from the centre of mass of the system are r_(1)andr_(2) respectively, then show that r_(1)=r((m_(2))/(m_(1)+m_(2)))andr_(2)=r((m_(1))/(m_(1)+m_(2)))

f(n)=sum_(r=1)^(n) [r^(2)(""^(n)C_(r)-""^(n)C_(r-1))+(2r+1)(""^(n)C_(r ))] , then