On day one, there are 100 particles in a laboratory experiment. On day n, where n≥2, one out of every n particles produces another particle. If the total number of particles in the laboratory experiment increases to 1000 on day m, then m equals
Question:
On day one, there are 100 particles in a laboratory experiment. On day n, where n≥2, one out of every n particles produces another particle. If the total number of particles in the laboratory experiment increases to 1000 on day m, then m equals
On day one, there are 100 particles in a laboratory experiment. On day n, where n≥2, one out of every n particles produces another particle. If the total number of particles in the laboratory experiment increases to 1000 on day m, then m equals
Options
Answer: (1) 19
Explanation:
Day-1: 100 particles. On day n (n ≥ 2), one out of every n particles produces another particle. So number of particles gets multiplied each day by: 1 + 1/n Total particles on day m: P(m) = 100 × (3/2) × (4/3) × (5/4) × ... × ((m+1)/m) This is a telescoping product. Everything cancels except first denominator and last numerator: P(m) = 100 × (m+1)/2 We are given P(m) = 1000, so: 100 × (m+1)/2 = 1000 (m+1)/2 = 10 m+1 = 20 m = 19 Answer: 19
Explanation:
Day-1: 100 particles. On day n (n ≥ 2), one out of every n particles produces another particle. So number of particles gets multiplied each day by: 1 + 1/n Total particles on day m: P(m) = 100 × (3/2) × (4/3) × (5/4) × ... × ((m+1)/m) This is a telescoping product. Everything cancels except first denominator and last numerator: P(m) = 100 × (m+1)/2 We are given P(m) = 1000, so: 100 × (m+1)/2 = 1000 (m+1)/2 = 10 m+1 = 20 m = 19 Answer: 19
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