Let a1, b1 be two real numbers such that 0 < a1 < b1. For n ≥ 1, we define an+1 = (anbn)^1/2 and bn+1 =(an + bn)/2 1) Prove that the sequence {an}n∈N is monotonically increasing and that the sequence {bn}n∈N is monotonically decreasing. 2) Show that the sequences {an} and {bn} are bounded. 3) Deduce that the two sequences converge and prove that they converge to the same limit. .
Let a1, b1 be two real numbers such that 0 < a1 < b1. For n ≥ 1, we
define an+1 = (anbn)^1/2 and bn+1 =(an + bn)/2
1) Prove that the sequence {an}n∈N is monotonically increasing and that the sequence {bn}n∈N is monotonically decreasing.
2) Show that the sequences {an} and {bn} are bounded.
3) Deduce that the two sequences converge and prove that they converge to the same limit.
.
Let a1 , b1 be two real numbers with 0 < a1 < b1 .
For n ≥ 1, we define
an+1 = (anbn)^1/2 and bn+1 = (an + bn)/2 .
Since, A.M. ≥ G.M. so,
bn+1 = (an + bn)/2 ≥ (anbn)^1/2 = an+1 or
bn+1 ≥ an+1 for all n.
So, it implies that
an+1 = (anbn)^1/2 ≥ (anan)^1/2 = an
or, an+1 ≥ an .
bn = (bn + bn)/2 ≥ (an + bn)/2 = bn+1
or, bn ≥ bn+1 .
Thus, we proved that for all n∈ N , { an } is a monotonically increasing sequence and { bn } is a monotonically decreasing sequence.
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