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?

The expression for the number of tiles at the n^th stage is 2n – 1.

For the 15th stage:

Substituting n = 15:

= 2(15) – 1

= 30 – 1

= 29

So, the 15th stage has 29 tiles.

For the 26th stage:

Substituting n = 26:

= 2(26) – 1

= 52 – 1

= 51

So, the 26th stage has 51 tiles.

To find which stage contains 21 tiles:

2n – 1 = 21

⇒ 2n = 21 + 1

⇒ 2n = 22

⇒ n = $\dfrac{22}{2}$

⇒ n = 11

So, the 11th stage contains 21 tiles.

To find which stage contains 47 tiles:

2n – 1 = 47

⇒ 2n = 47 + 1

⇒ 2n = 48

⇒ n = $\dfrac{48}{2}$

⇒ n = 24

So, the 24th stage contains 47 tiles.

Hence, the 15th stage has 29 tiles, the 26th stage has 51 tiles, the 11th stage contains 21 tiles, and the 24th stage contains 47 tiles.