Let T1 and T2 be the time periods of springs A and B when mass M is suspended from one end of each spring. If both springs are taken in series and the same mass M is suspended from the series combination, the time period is T, then:

Time period T = 2π√(M/K) => K = 4π²M/T²
Thus spring constant for first spring K1 = 4π²M/T1²(1)
Spring constant for second spring K2 = 4π²M/T2² (2)
For the series combination 1/Ks = 1/K1 + 1/K2
Let the time period for the series combination be T
=> T²/(4π²M) = T1²/(4π²M) + T2²/(4π²M) (using (1) and (2))
=> T² = T1² + T2²