A child arranges marbles in rows so that the first row has 1 marble, the second has 2 marbles, the third has 3, and so on up to 25 rows. How many marbles does the child use in all?

Step-by-step solution

Idea: The total is the sum of the first 25 natural numbers, S_n = n(n + 1)2 with n = 25.

1. Total = 1 + 2 + 3 + … + 25 = S₂₅.½ mark
2. S₂₅ = 25 × 262 = 25 × 13 = 325.1 mark
3. The child uses 325 marbles.½ mark

Check: Pairing: 12 pairs of (1 + 25) = 12 × 26 = 312, plus the middle row 13: 312 + 13 = 325 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Answering 25 (the number of rows) or 25 × 25 = 625.
• Forgetting to halve: 25 × 26 = 650.
• Using 25 × 24 ÷ 2 = 300 (the formula is n(n + 1) ÷ 2).

How this can come in the exam

Try one yourself

Logs are stacked with 1 log on top, 2 below it, 3 below that, …, and 14 in the bottom row. How many logs are there?

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