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When a spring is extended or compressed from its equilibrium length, the change in length is related to the force applied by the relation F = -kx, where F is the force applied, k is the spring constant and x is the change in length.
If two springs with a spring constant k1 and k2 are connected in parallel, when a force F is applied, the change in length is x. We get F = x*(k1 + k2) = keq*x. The equivalent spring constant keq = k1 + k2.
If two springs are connected in series, and a force F is applied, the force acts on both the springs changing the length of each differently. Let the spring constants of the springs be k1 and k2 and the change in length be x1 and x2. We have x1 = F/k1 and x2 = F/k2 and the total change in length x = F/ keq
=> x = x1 + x2
=> F/keq = F/k1 + F/k2
=> 1/keq = 1/k1 + 1/k2
=> keq = k1*k2 / (k1 + k2)
Therefore when two springs are connected in series the equivalent spring constant is k1*k2/ (k1 + k2). When two springs are connected in series the spring constant is k1 + k2.
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