An expert question on clever sums for P5–P6, with a step-by-step solution, diagrams and the answer. The first few questions of a competition paper are often clever sums, and the Hua Xia Cup has them in every year from P1 to P6.
The question
Work this out the clever way.
2 × 3 + 3 × 4 + 4 × 5 + … + 14 × 15 = ?
What is the answer?
Show the hints (3)
💡Hint 1: Try multiplying each term by 3: can it become the difference of two products of three consecutive numbers? For example 2 × 3 × 3 = 2 × 3 × 4 − 1 × 2 × 3.
💡Hint 2: k × (k + 1) × 3 = k × (k + 1) × (k + 2) − (k − 1) × k × (k + 1). Write every term this way and add: the middle ones cancel.
💡Hint 3: After cancelling, 3 × the sum = 14 × 15 × 16 − 1 × 2 × 3.
Worked solution
Have a go first, then read the solution step by step.
1.Times 3, each term: k × (k + 1) × 3 = k × (k + 1) × (k + 2) − (k − 1) × k × (k + 1). For example 2 × 3 × 3 = 2 × 3 × 4 − 1 × 2 × 3.
2.Write 2 × 3 to 14 × 15 this way: each term's first product cancels the next term's second.
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