An expert question on fraction problems for P6, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P5 and P6 qualifying rounds and finals; harder ones work backwards through fractions, or need clever shortcuts with fractions.
The question
Work out the sum below and write it as a fraction m/n in lowest terms (m and n are whole numbers with no common factor except 1).
3/4 + 5/36 + 7/144 + 9/400 + … + 13/1764 = ?
What is m + n?
Show the hints (3)
💡Hint 1: Look at the denominators 4, 36, 144, 400, …: are they products of squares? And how are the numerators 3, 5, 7, 9, … related to square numbers?
💡Hint 2: 4 − 1 = 3, 9 − 4 = 5, … each term = ((k + 1)² − k²)/(k² × (k + 1)²). Can it split into a difference of two fractions?
💡Hint 3: Each term = 1/k² − 1/(k + 1)². Split every term this way and the middle ones cancel; find k in the last term, subtract the two fractions left, and reduce.
Worked solution
Have a go first, then read the solution step by step.
2.The last term 13/1764: 1764 = 36 × 49 and 13 = 49 − 36, so k = 6; k runs from 1 to 6.
3.Each term = ((k + 1)² − k²)/(k² × (k + 1)²) = 1/k² − 1/(k + 1)²; for example 3/4 = 1/1 − 1/4.
4.Adding, the middle cancels: the sum = 1/1 − 1/49 = 48/49.
5.m + n = 48 + 49 = 97.
⚠️Common mistake: thinking 1/1 − 1/36 is left, when the last term brings 1/49; or missing that each numerator 2k + 1 is the gap between neighbouring squares.
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