Question

One bacterium splits into eight bacteria of the next generation. But due to environmental condition only 50% survives and the remaining 50% dies after producing the next generation. If the seventh generation number is 4,096 million, what is the number in the first generation?

• A. 1 million
• B. 2 million
• C. 4 million
• D. 8 million

Hint:

For exponential growth problems, use the relationship between generations and their respective factors: - The number of bacteria can be calculated using multiplication and division by powers of 8 and halving every generation.

Answer 1

The Correct Option is A

Let the number of bacteria in the first generation be \( N \). The bacteria split into 8 bacteria of the next generation, but only 50% survive. So, the number of bacteria in the next generation will be: \[ \text{Next Generation} = 8 \times 0.5 \times \text{Current Generation}. \] This means the number of bacteria decreases by a factor of 4 in each generation because of the 50% survival rate after each split. The population can thus be represented as: \[ N_{n} = N_{1} \times 8^n \times (0.5)^n, \] where \( N_{n} \) is the number of bacteria in the \(n\)-th generation. Given that the number of bacteria in the 7th generation is 4,096 million: \[ N_{7} = 4,096 \, \text{million} = N_1 \times 8^7 \times (0.5)^7. \] Now, \( 8^7 = 2^{21} \), and \( (0.5)^7 = 2^{-7} \), so: \[ 4,096 = N_1 \times 2^{21-7} = N_1 \times 2^{14}. \] We know that \( 2^{14} = 16,384 \), so: \[ 4,096 = N_1 \times 16,384. \] Solving for \( N_1 \): \[ N_1 = \frac{4,096}{16,384} = 0.25 \, \text{million}. \] This represents the number of bacteria in the first generation.