Let us suppose we have two springs with spring constant as k.
let us say we name the spring as S1 and S2.
Now let us connect the spring in series with S1 on left and S2 on right.
Now suppose the end of left S1 is fised to a wall and at the end of S2 on the right a block of mass M is attached.
Let us say the effective spring constant in this case becomes K.
Now let us say mass m is pulled to the right by a distance x and released
at the instant after releasing, let us say the spring 1 has been stretched to a distance x1 and spring 2 has been streched to a distance x2.
Now total strech given
x = x1+x2
Now consider a point at the junction of spring 1 and 2.
For this point, left hand force of the spring must be same as the right hand spring force.
so, kx1 = kx2
this means x1=x2
Now force acitng on the block is given by F=kx2
Now if there had been only 1 spring , then force would have been given by
or, Kx= kx/2 ....... putting the value of x2 from above in terms of x
cancelling x on both sides we get
This is the effective spring constant.
So time period is given by