Answer:
1) (Un) is a geometric sequence.
2) Un+1 - Un ≤ 0.
3) (Un) is decreasing.
4) (Un) is convergent.
5) sum = 155
Step-by-step explanation:
Un+1 = 0.5Un
U₀ = 80
1) U is a geometric sequence (reason : Un+1 = 0.5Un).
2) Un+1 - Un = 0.5Un - Un = Un×(0.5 - 1) = -0.5Un ≤ 0 (because Un ≥ 0).
Remark: use Mathematical induction to prove that Un ≥ 0
3) Un+1 - Un ≤ 0
then
Un+1 ≤ Un
then
the sequence U is decreasing.
4) U is decreasing and Un ≥ 0
Then U is convergent.
5) Since U is geometric sequence with first term U0 = 80 and common ratio = 0.5 .
Important: review the sum of consecutive terms of a geometric sequence.
Then
[tex]U_0+ U_1+ U_2+ U_3+ U_4=U_{0}\times \frac{1-\left( 0.5\right)^{5} }{1-0.5}=160\times(1-\left( 0.5\right)^{5} )=155[/tex]
