Arithmetic sequences

At a dorm, students play with rolls of toilet paper, building a pyramid of it. There is one roll at the top of the pyramid, in each lower row there is one roll more than in the one above it. How high will the pyramid be, given the fact that they have a total of $171$ rolls and one roll is $9.5\mathrm{cm}$ high?

A cyclist plans to travel $1666\mathrm{km}$ in $14$ days during her vacation. She knows that the number of kilometers she rides every day will decrease by the same number so she planned her route accordingly. At the beginning of the last day she was only $80\mathrm{km}$ away from her goal. What is the difference between the numbers of kilometers she rode on two consecutive days?

In Duolingo application, every user earns so-called lingots (=virtual currency), providing he or she learns at least $10$ minutes for $10$ consecutive days. For the first $10$ days a user earns $1$ lingot, $2$ lingots for the next ten-day period, $3$ lingots for the following $10$ days (i.e. for the first $30$ days a user earns $6$ lingots), etc. Find the smallest number of days in which a user can accumulate $1000$ lingots.

A certain type of bamboo grows $1.3\mathrm{m}$ per day during the vegetation period. Concerning the fact that it reached a height of $30\mathrm{m}$ after twenty days of regular growth, how tall was it at the beginning of the first day?