hi feyomi,

if the formula represent probabilities, then the sum of all probabilities

must equal 1.

These are sums to infinity.

For (i)

n=1, 2, 3....

the sum of these "probabilities" is

which is a geometric series, first term 2/3, common ratio 1/3, whose sum to infinity is

All probabilities sum to 1, hence the function g(n) is valid.

For (ii), use the hint

Hence, the sum is

As this is 1, this function is also valid.