Show that when a number of resistance are connected in parallel their equivalent resistance is smaller than the smallest of them .

Explanation

Suppose R be the equivalent resistance of the resistances `R_1,R_2,R_3,……….` connected in a parallel combination.
`therefore1/R=1/R_1+1/R_2+1/R_3+.....`........(1)
Suppose `R_1` is the smallest among the resistances connected in parallel.
Now, `1/R_1gt0,1/R_2gt0,1/R_3gt0,..........(2)`
So from equation (1) it can be said that
`1/Rgt1/R_1,1/Rgt1/R_2,1/Rgt1/R_3.....`
or, `RltR_1,RltR_2,RltR_3`
i.e., the equivalent resistance R is smaller than even the smallest resistance `R_1` of the parallel combination.