Growing figures

Identical matchsticks are laid to make a row of hexagons. Figure 1 uses 6 matchsticks, figure 2 uses 11 and figure 3 uses 16. One figure in this pattern uses 96 matchsticks. Which figure is it? Try it first. Pause if you like. Figure 1 is 1 hexagon, and it uses 6 matchsticks. In figure 2 the 2 hexagons share a matchstick. That shared one is counted once, so figure 2 uses 11. Figure 3 uses 16. 11 minus 6 is 5. 16 minus 11 is 5 as well. The count goes up by 5 each time. Now work back from 96. Figure 1 already holds 6, so 96 minus 6 is 90 matchsticks added after it. Each gap adds 5, so 90 divided by 5 is 18 gaps. Figure 1 comes before the first gap. 18 gaps after figure 1 lands on figure 19. So figure 19 uses 96 matchsticks.

Figure 1 is a T-shape made of 4 identical squares and is 12 centimetres tall. Figure 2 is made by adding one square to the end of each of the three arms of figure 1. Every figure after that is made from the one before it in the same way. Find the area of figure 20. Try it first. Pause if you like. The only length given is figure 1's height. The T is 2 squares tall, so 12 divided by 2 is 6 centimetres for one square's side. 6 times 6 is 36 square centimetres for one square. Now count squares. Figure 1 has 4, figure 2 has 7, and 7 minus 4 is 3. Figure 3 has 10, and 10 minus 7 is 3 too: one new square on each of the three arms. Figure 1 is already there, so reaching figure 20 takes 20 minus 1, which is 19 gaps. Each gap adds 3, so 19 times 3 is 57 squares added. 4 plus 57 is 61 squares in figure 20. A count is not an area, so the two meet only now. 61 squares of 36 square centimetres each is two thousand, one hundred and ninety six square centimetres. So the area of figure 20 is two thousand, one hundred and ninety six square centimetres. A growing pattern starts at the first figure and adds the same difference for every gap.