Your first equation can be extended to as many resistors as you like.

1/r = 1/r1 + 1/r2 + 1/r3 + ... + 1/rn

When you express that sum over a common denominator (r1*r2*r3*...*rn), you find the numerator becomes (for 3 resistors)

r2*r3 + r1*r3 + r1*r2

Inverting that makes your "product over sum" expression into

r = r1*r2*r3/(r2*r3 + r1*r3 + r1*r2)

Each element of the sum is the product with one term removed, just as it is in the 2-resistor case.

1/r = 1/3 + 1/3 + 1/6 = 5/6

r = 6/5

r = (3*3*6)/(3*6 + 3*6 + 3*3) = 54/(18+18+9) = 54/45 = 6/5

1/r = 1/r1 + 1/r2 + 1/r3 + ... + 1/rn

When you express that sum over a common denominator (r1*r2*r3*...*rn), you find the numerator becomes (for 3 resistors)

r2*r3 + r1*r3 + r1*r2

Inverting that makes your "product over sum" expression into

r = r1*r2*r3/(r2*r3 + r1*r3 + r1*r2)

Each element of the sum is the product with one term removed, just as it is in the 2-resistor case.

1/r = 1/3 + 1/3 + 1/6 = 5/6

r = 6/5

r = (3*3*6)/(3*6 + 3*6 + 3*3) = 54/(18+18+9) = 54/45 = 6/5