How do you understand the last 3 steps for convolution in the z-transform?

[Toyona10]

Greetings,
We just had 1.5 classes about this topic (due to some unavoidable circumstances in our country...) so our teacher just scribbled and roughly explained them on the board...So after i got home i couldn't get some steps he did for convolution in z-transform:

find inverse z-transform of z^2/(z-2)(z-5)

so here, inverse of z/(z-2) is 2^n which is Un and inverse of z/(z-5) is 5^n which is Vn
so Vn*Un = Ʃ(from m=0 to n) U_n-mV_m

=Ʃ(from m=0 to n)2^n-m5^m
=Ʃ(from m=0 to n)2ⁿ(5^m/2^m
=2^nƩ(from m=0 to n)[5/2]^m
=2^n[1+(5/2)+(5/2)^2 + (5/2)^3...(5/2)^n]

By convolution-

=2^n{[1(5/2)ⁿ⁺¹ - 1]/(5/2 -1)}
=2^n[(5ⁿ⁺¹-2ⁿ⁺¹)/2ⁿ⁺¹×3/2]
=1/3[5ⁿ⁺¹-2ⁿ⁺¹]

I don't get the last 3 steps, I would REALLY appreciate it if somebody would explain that ^^

 

[Toyona10]

anyone??