What is the condition for a body suspended at the end of a spring to undergo simple harmonic oscillation?

$A$ mass $M$ is suspended from a light spring. An additional mass $m$ added displaces the spring further by a distance $x$. Now the combined mass will oscillate on the spring with period:

Two identical particles each of mass $m$ are interconnected by a light spring of stiffness $k$. The time period for small oscillations is equal to:

$A$ mass $m$ is suspended separately by two springs of spring constants $k_1$ and $k_2$. The time periods of oscillations in the two cases are $T_1$ and $T_2$ respectively. If the same mass $m$ is suspended by connecting the two springs in parallel (as shown in the figure),then the time period of the oscillation is $T$. The correct relation is:

$A$ mass $m$ is suspended separately by two different springs of spring constant $K_1$ and $K_2$ giving the time periods $t_1$ and $t_2$ respectively. If the same mass $m$ is connected by both springs as shown in the figure,then the time period $t$ is given by the relation:

Show that the oscillations due to a spring are simple harmonic oscillations and obtain the expression for the period of oscillation.