91Ó°ÊÓ

Write each rational number as the quotient of two integers in simplest form. $$0 . \overline{63}$$

Write each rational number as the quotient of two integers in simplest form. $$0 . \overline{123}$$

Write each rational number as the quotient of two integers in simplest form. $$0 . \overline{422}$$

Write each rational number as the quotient of two integers in simplest form. $$0 . \overline{355}$$

Newton's approximation to the square root of a number, \(N\), is given by the recursive sequence $$a_{1}=\frac{N}{2} \quad a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{N}{a_{n-1}}\right)$$ Approximate \(\sqrt{7}\) by computing \(a_{4} .\) Compare this result with the calculator value of \(\sqrt{7} \approx 2.6457513\)

If \(f(x)\) is a linear polynomial, show that \(f(n),\) where \(n\) is a positive integer, is an arithmetic sequence.

Find the formula for \(a_{n}\) in terms of \(a_{1}\) and \(n\) for the sequence that is defined recursively by \(a_{1}=3\) \(a_{n}=a_{n-1}+5\)

Find a formula for \(a_{n}\) in terms of \(a_{1}\) and \(n\) for the sequence that is defined recursively by \(a_{1}=4, a_{n}=a_{n-1}-3\)

Suppose \(a_{n}\) and \(b_{n}\) are two sequences such that \(a_{1}=4\) \(a_{n}=b_{n-1}+5\) and \(b_{1}=2, b_{n}=a_{n-1}+1 .\) Show that \(a_{n}\) and \(b_{n}\) are arithmetic sequences. Find \(a_{100}\)

If the sequence \(a_{n}\) is a geometric sequence, make a conjecture about the sequence \(\log a_{n}\) and give a proof.