You draw a sea of reds, with blues on the i-th, j-th, and n-th (terminating) draws

1<=i < j < n

Ex: i = 4, j = 10, n = 15: R R R B R R R R R B R R R R B

The probabilitiy of such an string, is:

.

Therefore .

Thus

where .

Therefore

Have .

Thus

Therefore .

Therefore

But .

So .

Thus

Conclude that the expectation is .

(I'm going to double check this later - it took too long. However, it looks like a reasonable result.)