Dynamic programming
The value of the goal function is
VT (t+ 1 )(w)=
r w
∑tn+=^10 kn
+
s t
∑
n= 0
kn
s
k
w w
∑tn+=^10 kn
=
=
p
w(q^1
∑tn+=^10 kn
+
s
t
∑
n= 0
kn
s
k∑
t+ 1
n= 0 kn ^1
∑tn+=^10 kn
)=
=
p
w(q^1
∑tn+=^10 kn
+
s
t
∑
n= 0
kn
s
k∑
t+ 1
n= 1 kn
∑tn+=^10 kn
)=
=
p
w(q^1
∑tn+=^10 kn
+
s
t+ 1
∑
n= 1
kn
s
∑tn+=^11 kn
∑tn+=^10 kn
)=
p
wq^1
∑tn+=^10 kn
( 1 +
t+ 1
∑
n= 1
kn)=
p
w
s
t+ 1
∑
n= 0
kn.