Grade 12 Sequences, I don't understand the exercise at all I am lost

[BenZad]

Hello everyone, I am lost with this exercise and I don't know how to solve it.

 

[Dr.Peterson]

Hello everyone, I am lost with this exercise and I don't know how to solve it.

Presumably you can answer the first question.

To proceed, I would start by "playing" with it (experimenting with small numbers). Start with a 1x3 rectangle. How many 'legal' ways are there to tile it? (That's question 2a.) Then move to 2x3, then 3x3, and look for a pattern that suggests that 2b is correct. While you do the counting, think how you might use one result to determine the next. That may lead you in the direction of an inductive proof for 2b. Once you have a formula, you can answer 2c.

If you are stuck at any point, write back and show us what you've done.

 

[HallsofIvy]

The hall is 20 "squares" long by 3 "squares" wide so requires a total of 3(60)= 180 squares. Since any one square can be any of 3 colors, there are \(\displaystyle 3^{180}\) different patterns.