Question

when the resistance are converted and parallel, the equivalent resistance is less than each of the resistance. true or false​

Answer 1

Explanation:

true

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Answer 2

Answer:

true

4

We can prove it by induction. Let

1R(n)eq=1R1+⋯+1Rn

Now, when n=2, we find

1R(2)eq=1R1+1R2⟹R(2)eq=R1R2R1+R2=R11+R1R2=R21+R2R1

Since R1R2>0, we see that R(2)eq<R1 and R(2)eq<R2 or equivalently R(2)eq<min(R1,R2).

Now, suppose it is true that R(n)eq<min(R1,⋯,Rn). Then, consider

1R(n+1)eq=1R1+⋯+1Rn+1Rn+1=1R(n)eq+1Rn+1

Using the result from n=2, we find

R(n+1)eq<min(Rn+1,R(n)eq)<min(Rn+1,min(R1,⋯,Rn))

But

min(Rn+1,min(R1,⋯,Rn))=min(Rn+1,R1,⋯,Rn)

Therefore

R(n+1)eq<min(R1,⋯,Rn,Rn+1)

Thus, we have shown that the above relation holds for n=2, and further that whenever it holds for n, it also holds for n+1. Thus, by induction, it is true for all n≥2.