Question

two light spring of spring constant K1 and K2 are connected 1) in series and 2) in parallel , what is the equivalent spring constant of the combination in each case.

Answer 1

Equivalent spring constant

Keq = K1+K2 (In parallel)

1/Keq= 1/K1+1/K2 (In series)

Answer 2

in series combination , same force acts on each spring but total elongation is the sum of elongation by each spring.
we know from Hook's law , $F=-Kx$
Let $x_1$ and $x_2$ are elongation of springs. then,
$x_1+x_2=x$
$\frac{F}{K_1}+\frac{F}{K_2}=\frac{F}{K_{eq}}$
e.g., $K_{eq}=\frac{K_1K_2}{K_1+K_2}$

in parallel combination , elongation of springs are same. but net force is the sum of force due to each spring.
e.g., $F_1+F_2=F_{net}$
$K_1x+K_2x=K_{eq}x$
hence, $K_{eq}=K_1+K_2$