Math patterns: table

So let's say I have tables where I can fit one person at either of the short ends of the table. So I could fit one person there. I could fit one person there. You could view this as we're looking from above the table here. So we could put one person at either of the short ends of the table. And then on these longer ends right over here, we can fit two people. We can fit two people at the longer end. So when you have one table, you could fit one, two, three, four, five, six people. You could fit six people. Now let's think about what happens as we add tables end to end to this table right over here. So let's imagine now two tables. So here we have one table, and it's going to touch ends with this table right over here. And because it touches ends right over here-- we're making it one big continuous table-- you can't fit someone here anymore. So now how many people can we fit? Let's see. We can fit one, two, three, four, five. And then on this table, which is identical, you could fit six, seven, eight, nine. And then you could put one person at the end right over here. So when you have two tables end to end, you can fit a total of 10 people. Let's keep going and see if we can think of a pattern here. So let's put three tables here-- so one table, two tables, and three tables. So just as before, we could put one at each end. So that's two people. Then we have 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 folks-- 14 folks. So what does it look like is happening here. Well if you just look at the numbers, we went from 6 to 10 to 14. It looks like we're adding four people every time we actually add a table. Now does that actually makes sense? So let's think about this first situation. Let's imagine these are real people, and I'll make this person in blue right over here. If you were to bring over this new table, if you bring over table two-- so this is table one-- this blue person has to move. And so where could they move? Let's say that they always insist on sitting at the end of a table. So the blue person moves to the new end of the table. They move right over here. So how many new people could move to this combined table now that you brought this second table in? Well the new people I will do in this purple color. The new people are that person-- let me do it in a more unique color-- this person, this person, this person, and this person. So you were able to add four new people with the new table. One way to think about it is a new table is going to have one usable end here. That usable end is going to be taken by the person who was already at the usable end of when you had less tables. And so the real addition is the two sides here. So you're adding four people every time you add a table. So it makes complete sense. So based on this, you could think about, without even having to draw these diagrams, how many people you would be able to fit if you had four or five or six or however many tables. So you could imagine, if you have four tables, we just have to add four, and you should be able to sit 18 people. If you have five tables, you should be able to fit 22 people and on and on and on.