What is the value of equivalent spring constant if two spring of spring constant k and 2k are connected in parallel?

The math has been explained very well, but you said that you already understood the math, and it was just the apparent conceptual contradiction about energy transfer that bothered you.

The problem is very subtle and interesting. If you put your "gut reaction" into words, it would probably say that since the 15N force supported is the same for both spring arrangements, they do the same amount of work.   But at the same time, your physics-trained mind knows better.   It knows that work, or energy transfer, only occurs when force is exerted through a distance.

When you hold a book motionless at arm's length out in front of you, no work is done on the book.   Yet after a minute, your body feels like it's doing a lot of work, that energy is being transferred, and it is--in your muscle cells.   So your body is not wrong.   But no energy is being transferred to the book.

These "common sense" things that our bodies and senses tell us are about the world around us are reinforced thousands of times by everyday experiences before we study physics.   They are hard to get rid of.   Aristotle's ideas that heavy things fall faster than light things, and that the force of the hand is still on a rock after you throw it, were believed by many for about 2000 years before Galileo stepped in.

Answer

Hint: Two massless springs that follow Hooke's Law are said to be connected in parallel when they are connected by a thin, vertical rod, as shown in the diagram below. The formula for capacitors connected in parallel in an electrical circuit can be used to find the value of k.

Complete answer:

Note: When two or more springs are connected end-to-end or point-to-point in mechanics, they are said to be in series, and when they are connected side-by-side, they are said to be in parallel; in both cases, they act as a single spring.

Parallel.

When two massless springs following Hooke's Law, are connected via a thin, vertical rod as shown in the figure below, these are said to be connected in parallel. Spring 1 and 2 have spring constants #k_1# and #k_2# respectively. A constant force #vecF# is exerted on the rod so that remains perpendicular to the direction of the force. So that the springs are extended by the same amount. Alternatively, the direction of force could be reversed so that the springs are compressed.

#k=k_1+k_2#

Series.

When same springs are connected as shown in the figure below, these are said to be connected in series. A constant force #vecF# is applied on spring 2. So that the springs are extended and the total extension of the combination is the sum of elongation of each spring. Alternatively, the direction of force could be reversed so that the springs are compressed.

This system of two springs in series is equivalent to a single spring, of spring constant #k#. The value of #k# can be found from the formula that applies to capacitors connected in series in an electrical circuit.

For spring 1, from Hooke's Law

#F=k_1x_1#

where #x_1# is the deformation of spring.

Similarly if #x_2# is the deformation of spring 2 we have

#F=k_2x_2#

Total deformation of the system

#x_1+x_2=F/k_1+F/k_2#
#=>x_1+x_2=F(1/k_1+1/k_2)#

Rewriting and comparing with Hooke's law we get

#k=(1/k_1+1/k_2)^-1#