Periodic time of oscillation `T_(1)` is obtained when a mass is suspended from a spring and if another spring is used with same mass then periodic time of oscillation is `T_(2)` . Now if this mass is suspended from series combination of above springs then calculated the time period.

Explanation

`T_(1) = 2pi sqrt((m)/(k_(1))) rArr T_(1)^(2) = 4pi^(2) (m)/(k_(1)) rArr k_(1) = (4pi^(2)m)/T_(1)^(2)` and
`T_2 = 2pisqrt((m)/(k_(2))) rArr T_(2)^(2) = 4pi^(2)m/k_(2) rArr k_(2) = -(4pi^(2)m)/(T_(2)^(2))`
Now `T = 2pisqrt((m)/(k'))` where `1/k' = 1/k_(1) + (1)/(k_(2)) rArr k' = (k_(1)k_(2))/(k_(1) + k_(2)) = (((4pi^(2)m)/(T_(1)^(2)))((4pi^(2)m)/(T_(2)^(2))))/((4pi^(2)m)/(T_(1)^(2)) + (4pi^(2)m)/(T_(g)^(2)))`
`k' = (4pi^(2)m[(4pi^(2)m)/(T_(1)^(2)+T_(2)^(2))])/(4pi^(2)m[1/(T_(1)^(2)) + 1/(T_(2)^(2))])=(4pi^(2)m)/(T_(1)^(2)+T_(2)^(2))`
`:. T = 2pisqrt((m)/(k')) = 2pisqrt((m)/((4pi^(2)m)/(T_(1)^(2) + T_(2)^(2))))= sqrt(T_(1)^(2) + T_(2)^(2))`