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Fraction sequences

Solve the first problem 14 problems

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1. Next term of a fraction sequence ●○○○○ 10 ptsFind the next term of the sequence $0, 0.5, 0.\overline{6}, 0.75, \dots$ and give it as a decimal. Round to three decimal places.
2. Hundredth term of a fraction sequence ●○○○○ 10 ptsThe sequence $\frac{1}{2}, \frac{3}{5}, \frac{5}{8}, \frac{7}{11}, \frac{9}{14}, \ldots$ follows a pattern. Find the 100th term of the sequence. Round your answer to two decimal places.
3. Missing term in a fraction sequence ●○○○○ 10 ptsFind the missing term in the sequence $1, \frac{3}{2}, \_\_\_, \frac{7}{8}, \frac{9}{16}$. Give your answer as a decimal.
4. Sixth term of a fraction sequence ●●○○○ 20 ptsFind the 6th term of the sequence $\frac{1}{3}, \frac{3}{5}, \frac{5}{8}, \frac{7}{12}, \frac{9}{17}, \ldots$. Round your answer to two decimal places.
5. Doubling map sequence term ●●○○○ 20 ptsA sequence $\{a_n\}$ satisfies $a_{n+1}=2a_n$ if $0 \leqslant a_n < \frac{1}{2}$ and $a_{n+1}=2a_n-1$ if $\frac{1}{2} \leqslant a_n < 1$, for every positive integer $n$, and $a_1=\frac{6}{7}$. Find $a_{24}$. Round to two decimal places.
6. Twentieth term of a radical sequence ●●○○○ 20 ptsConsider the sequence $\sqrt{2}$, $-2$, $\sqrt{6}$, $-2\sqrt{2}$, $\sqrt{10}$, $-2\sqrt{3}$, $\ldots$. Following this pattern, find the 20th number in the sequence. Round to two decimal places.
7. Counting nine-term sequences with ratio constraints ●●●○○ 50 ptsThe sequence $\{a_n\}$ has nine terms, $a_1=a_9=1$, and for each $i\in\{1,2,\ldots,8\}$ we have $\frac{a_{i+1}}{a_i}\in\left\{2,1,-\frac{1}{2}\right\}$. How many such sequences are there?
8. Position of a fraction in a sequence ●●●○○ 50 ptsThe positive rational numbers are arranged in the order $\frac{1}{1}, \frac{2}{1}, \frac{1}{2}, \frac{3}{1}, \frac{2}{2}, \frac{1}{3}, \frac{4}{1}, \frac{3}{2}, \frac{2}{3}, \frac{1}{4}, \cdots$. Find the position number of the number $\frac{1989}{1949}$ in this sequence.

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