This is my classmate's idea: mainly use conditional expectation.

Let EXn be the expectation of scoring N shots, then consider N shots. There are two situations:

First, the first miss, when E(Xn| first miss) = E(Xn- 1)

Second, hit the target for the first time, and then consider the second time.

If the second miss, then E(Xn| one miss and two misses) = E(Xn-2),

If the second time, there is E(Xn| one middle school and two middle schools) = E(Xn-2)+ 1.

So according to the conditional expectation formula,

EXn =( 1-p)E(Xn- 1)+p( 1-p)E(Xn-2)+PP[E(Xn-2)+ 1]=( 1-p)E(Xn- 1)+pE(Xn-2)+PP

This is a recursive formula. It is easy to know the first two terms, EX 1 = 0, EX2 = pp, and then use the characteristic equation or do mathematical induction.