A lab experiment measures the number of organisms at 8 am every day. Starting with 2 organisms on the first day, the number of organisms on any day is equal to 3 more than twice the number on the previous day. If the number of organisms on the nth day exceeds one million, then the lowest possible value of n is
Explanation:
Number of micro-organisms on day 1 = 2
Number of micro-organisms on day 2 = 2 × 2 + 3 = 22 + 3 Number of micro-organisms on day 3 = 2 × (22 + 3) + 3 = 23 + 3 × (2 + 1) Number of micro-organisms on day 4 = 2 × (23 + 3 × (2 + 1)) + 3 = 24 + 3 × (22 + 2 + 1) Number of micro-organisms on day 5 = 2 × (24 + 3 × (22 + 2 + 1)) + 3 = 25 + 3 × (23 + 22 + 2 + 1)
∴ Number of micro-organisms at the end of day n = 2n + 3 × (2n-2 + … + 22 + 2 + 1) = 2 × 2n-1 + 3 × (2n-1 - 1) = 5 × 2n-1 - 3
Now, 5 × 2n-1 - 3 ≥ 10,00,000 ⇒ 2n-1 ≥ 2,00,000 + 3/5
The least value of (n - 1) satisfying above inequality is 18. ⇒ n - 1 = 18 ⇒ n = 19
Hence, 19.
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