Evaluate 1×2 + 2×3 + 3×4 + … + 9×10.

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Sum of consecutive products

Mental mathHard

Evaluate $1 \times 2 + 2 \times 3 + 3 \times 4 + \dots + 9 \times 10$ .

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Hint

Look for a way to write each product so that neighbouring terms cancel.

Solution

Use $k (k + 1) = [k (k + 1) (k + 2) - (k - 1) k (k + 1)] \div 3$ .
Summing from $k = 1$ to $9$ , the terms telescope to $9 \times 10 \times 11 \div 3$ .
That is $990 \div 3$ .
So the sum is $330$ .

Answer: $330$