Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 A geometric sequence has all terms positive. Its 2nd term is 6 and its 4th term is 54. Which expression gives its nth term?
Show worked solution
r2 = 54 ÷ 6 = 9, so r = 3 and the first term is 6 ÷ 3 = 2. The nth term is 2 × 3n−1. Check: 2 × 3 = 6 ✓ and 2 × 27 = 54 ✓. 2 × 3n gives a 2nd term of 18 (one power too many); 6 × 3n−1 uses the 2nd term as the first; 24n − 42 treats the sequence as arithmetic.
2 Matchsticks make two rows of squares. Figures 1, 2 and 3 are shown, and the pattern continues.
Show worked solution
Figures 1, 2, 3 use 7, 12, 17 sticks: each new column adds 3 across and 2 down, 5 sticks. So Figure n uses 5n + 2. Figure 10: 5 × 10 + 2 = 52. 5n + 2 = 202 gives n = 40.
3 What is the units digit of 830 + 930?
Show worked solution
Powers of 8 end in 8, 4, 2, 6 (cycle of 4); 30 = 4 × 7 + 2, so 830 ends in 4. Powers of 9 end in 9, 1 (cycle of 2); 30 is even, so 930 ends in 1. 4 + 1 = 5.
Level 2 · Core
4 A row of 2025 coins all show heads. A move turns over exactly 2 coins (any 2 you like). Can you make all 2025 coins show tails?
Show worked solution
A move changes the number of heads by +2, 0 or −2, so it stays odd or stays even. It starts at 2025, odd, and all tails means 0 heads, even. No: the number of heads always stays odd. "1013 moves" pairs up 2024 coins and forgets the last coin cannot be turned alone; no number of moves can work, so "more than 2025 moves" is wrong too.
5 A theatre has a front row of 14 seats. Each row has 3 more seats than the row in front of it, and the back row has 71 seats.
Show worked solution
Rows: (71 − 14) ÷ 3 + 1 = 19 + 1 = 20. Total: 20 × (14 + 71) ÷ 2 = 10 × 85 = 850 seats. Forgetting the + 1 gives 19 rows, the fence-post error.
6 In this sequence each term after the 2nd is the sum of the two terms before it. The 5th term is 23 and the 7th term is 61. What is the 1st term?
Show worked solution
6th = 61 − 23 = 38. 4th = 38 − 23 = 15. 3rd = 23 − 15 = 8. 2nd = 15 − 8 = 7. 1st = 8 − 7 = 1. Check forwards: 1, 7, 8, 15, 23, 38, 61 ✓.
Level 3 · Challenge
7 Three statements about the sequence 4, 11, 18, 25, …:
1 Its 100th term is 697.
2 The sum of its first 100 terms is 70 100.
3 If its first 20 terms are written on a board and a move erases two numbers and writes their difference, then after 19 moves the last number left is odd.
Which statements are correct?
1 Its 100th term is 697.
2 The sum of its first 100 terms is 70 100.
3 If its first 20 terms are written on a board and a move erases two numbers and writes their difference, then after 19 moves the last number left is odd.
Which statements are correct?
Show worked solution
100th term = 4 + 99 × 7 = 697 (1 ✓). Sum = 100 × (4 + 697) ÷ 2 = 35 050; 70 100 forgets to halve (2 ✗). The first 20 terms add to 20 × (4 + 137) ÷ 2 = 1410, even. Replacing a and b by a − b changes the total by 2b, so the total stays even and the last number is even (3 ✗). 1 only.
8 The sum of 20 consecutive whole numbers is 830. Find the smallest and the largest of them.
Show worked solution
20 × (a + a + 19) ÷ 2 = 830, so 2a + 19 = 83 and a = 32. Smallest 32, largest 32 + 19 = 51. Check: 20 × (32 + 51) ÷ 2 = 10 × 83 = 830 ✓. Using a + 20 for the largest is the fence-post error.
9 Figure n of this dot pattern is a rectangle n dots tall and n + 1 dots wide. Figure k and the next figure, Figure k + 1, have 800 dots between them. Find k.
Show worked solution
k(k + 1) + (k + 1)(k + 2) = (k + 1)(2k + 2) = 2(k + 1)2. So 2(k + 1)2 = 800, (k + 1)2 = 400, k + 1 = 20 and k = 19. Check: 19 × 20 + 20 × 21 = 380 + 420 = 800 ✓.