Domestic bees make their honeycomb by starting with a single hexagonal cell, then forming ring after ring of hexagonal cells around the initial cell, as shown. The number of cells in successive rings forms an arithmetic sequence.

How many cells are in the honeycomb after the ninth ring is formed?

You are standing next to a bucket and are tasked with collecting $100$ potatoes, but can only carry one potato at a time. The potatoes are in a line in front of you, with the first potato $1$ meter away and each subsequent potato is located an additional one meter away.

How much distance would you cover while performing this task?

A small terrace at a football ground comprises of $15$ steps each of which is $50 \mathrm{m}$ long and built of solid concrete. Each step has a rise of $\frac{1}{4} \mathrm{m}$ and a tread of $\frac{1}{2} \mathrm{m}$ (See figure). Calculate the total volume (in ${\mathrm{m}}^3$) of concrete required to build the terrace.

In a bucket and ball race, a bucket is placed at the starting point, which is $5 \mathrm{m}$ from the first ball, and the other balls are placed $3 \mathrm{m}$ apart in a straight line. There are ten balls in the line.

A competitor starts from the bucket, picks up the nearest ball, runs back with it, drops it in the bucket, runs back to pick up the next ball, runs to the bucket to drop it in, and he continues in the same way until all the balls are in the bucket. What is the total distance the competitor has to run in $\mathrm{m}$.