Let the sum $ \sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)} $, written in the rational form be $ \frac{p}{q} $ (where $ p $ and $ q $ are co-prime), then the value of $ \left[\frac{q-p}{10}\right] $ is, (where [.] is the greatest integer function)

Question

Let the sum $ \sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)} $, written in the rational form be $ \frac{p}{q} $ (where $ p $ and $ q $ are co-prime), then the value of $ \left[\frac{q-p}{10}\right] $ is, (where [.] is the greatest integer function),

MARKS App · Accepted Answer

∑ n = 1 9 1 2 n + 2 - n n n + 1 n + 2 = 1 2 1 1.2 - 1 2.3 + 1 2.3 - 1 3.4 + … + 1 9 × 10 - 1 10 × 11 = 1 2 1 2 - 1 110 = 1 2 55 - 1 110 = 27 110 ⇒ q - p = 110 - 27 = 83

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