This leads us to conclude that the number of squares at Stage n is given by 2n − 1.
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?
Step-by-step solution
Idea: To find the tiles, put the stage number in place of n. To find the stage, set 2n − 1 equal to the number of tiles and solve the linear equation.
- Stage 15: 2(15) − 1 = 30 − 1 = 29 tiles.½ mark
- Stage 26: 2(26) − 1 = 52 − 1 = 51 tiles.½ mark
- 21 tiles: 2n − 1 = 21 ⇒ 2n = 22 ⇒ n = 11. Stage 11.½ mark
- 47 tiles: 2n − 1 = 47 ⇒ 2n = 48 ⇒ n = 24. Stage 24.½ mark
Check: Stage 11: 2(11) − 1 = 21 ✓. Stage 24: 48 − 1 = 47 ✓.
Answer to write in the exam
Tiles at Stage n = 2n − 1
Stage 15: 2(15) − 1 = 29 tiles
Stage 26: 2(26) − 1 = 51 tiles
2n − 1 = 21 ⇒ 2n = 22 ⇒ n = 11
2n − 1 = 47 ⇒ 2n = 48 ⇒ n = 24
∴ 29 tiles, 51 tiles; 21 tiles at Stage 11, 47 tiles at Stage 24
Common mistakes that cost marks
- Working out 2n − 1 as 2(n − 1): for n = 15 that gives 28 instead of 29. Multiply first, then subtract 1.
- Solving 2n − 1 = 21 as 2n = 20. Add 1 to both sides: 2n = 22.
- Answering “21 tiles” with the value of 2n − 1 at n = 21 (that is, 41). The question asks which stage, so solve for n.
How this can come in the exam
The nth term of a pattern is 3n + 2. Which term equals 62?
- 18th
- 20th
- 21st
- 22nd
Show answer
(B) 20th
3n + 2 = 62 ⇒ 3n = 60 ⇒ n = 20.
Assertion (A): No stage of the pattern 2n − 1 has 40 tiles.
Reason (R): 2n − 1 is always an odd number for a natural number n.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
2n is even, so 2n − 1 is odd; 40 is even, so it never appears. R explains A.
Try one yourself
For the pattern 2n − 1, how many tiles are in the 40th stage, and which stage has 99 tiles?
Show answer
2(40) − 1 = 79 tiles. 2n − 1 = 99 ⇒ n = 50: Stage 50.
More questions like this
- Bela has ₹100 for pocket money. She spends ₹5 every day. After how many days will she be left with ₹40?
- Bela has ₹100 for pocket money. She spends ₹5 every day. Observe that the amount left on the nth day will be ₹(100 − 5n).
What amount will be left on the 15th day? How many days will it take for the entire amount to be spent? - An auto-rikshaw fare starts at ₹25 and remains the same for the initial 2 km. Then it increases by ₹15 per km. What will be the fare for a travel of 10 km?
- Observe that the total fare for a travel of n km will be ₹25 + 15 × (n − 2) = 15n − 5, when n ≥ 2.
For how many km will the fare be ₹130? - A student has ₹500 in her savings bank account. She gets ₹150 every month as pocket money. How much money will she have at the end of every month from the second month onwards? Find a linear expression to represent the amount she will have in the nth month.