Exercise.

Prove that if _{
+1=
1+}_{
}
where _{10
}, then the sequence
(_{
)
} converges to the positive root of
^{2+=
}

NOTE: The word discuss

is used to frame deliberately open-ended
questions. Say as much as you reasonably can in the time available.
All your assertions should be proved. Discuss the convergence of
these sequences

clearly is asking for an indication of whether
the sequences do or do not converge, and if possible to what.

**Solution to Exercise 2(a)**
For ^{
12
(1+1+1
)
12
}
so the sequence is monotonic decreasing. Each value
(1+^{
12
} is greater than zero
because 1+_{
} for some ^{
12=1}