Geometric progression and the Binomial expansion

A sequence U1,U2,U3,.... is defined by U1=10, Un+1=0.9Un
(a) Find the value of U4
∞
(b) Find Σ Ur.
　　　r=1

(a)
Obviously, Un=(0.9n-1)10
Then U4=(0.9³)10=7.29

(b)
∞
Σ Ur.
r=1

=10+0.9(10)+0.9²(10)+......
=10/(1-0.9)
=100

此是一公比q=0.9的等比數列,可利用公式
U4=U1*q^3
=10*0.729=7.29

∞
Σ Ur=U1/(1-q)=10/0.1=100
r=1