Q1.
Using the expression 2n – 1, can you find out how many tiles will be there in the 15th stage and the 26th stage of the pattern? Also, which stage will contain 21 tiles and 47 tiles?
Answer
Forward — stage given, tiles wanted. Substitute in 2n − 1.
n = 15: 2(15) − 1 = 30 − 1 = 29 tiles
n = 26: 2(26) − 1 = 52 − 1 = 51 tiles
n = 26: 2(26) − 1 = 52 − 1 = 51 tiles
Backward — tiles given, stage wanted. Now 2n − 1 becomes a linear equation.
2n − 1 = 21 → 2n = 22 → n = 11
2n − 1 = 47 → 2n = 48 → n = 24
2n − 1 = 47 → 2n = 48 → n = 24
So Stage 11 has 21 tiles and Stage 24 has 47 tiles.
Why the rule can be run both ways: 2n − 1 is a machine that doubles and subtracts 1. To reverse it you undo those steps in the opposite order — add 1, then halve. Because doubling and halving are exact inverses, each tile-count comes from one and only one stage; there is never a choice. This is the same one-solution property that made the chess-club question have a single answer.
Check it yourself: 2n − 1 is always odd, so no stage can ever hold 20 or 50 tiles. Ask for 20 and the equation gives n = 10.5, which is not a stage number.