Question 43 of 51intermediate🔧 ApplyNumerical4 marks

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of the 2nd hour, 4th hour and nthn^{th} hour?

Correct Answer

(2nd hour) 120

(4th hour) 480

(n-th hour) 30 x 2n2^{n}

Exercise: END-OF-CHAPTER EXERCISES | Q: 7 | (Page 22)
For More Understanding

Explanation

The number of bacteria doubles every hour, which means each term of the sequence is obtained by multiplying the previous term by a constant multiplier known as the common ratio, which is 2. Starting with 30 bacteria originally, the sequence of bacteria at the end of each hour is 30, 30×230 \times 2, 30×230 \times 2², 30×230 \times 2³, and so on. Following this pattern, the number of bacteria at the end of the nth hour can be expressed as 30×230 \times 2ⁿ.

Solution Steps

  1. Initial count and common ratio: The original number of bacteria is 30, and the common ratio is 2 since the count doubles every hour.

  2. Bacteria at the end of the 2nd hour: 30×230 \times 2² = 30×430 \times 4 = 120.

  3. Bacteria at the end of the 4th hour: 30×230 \times 2⁴ = 30×1630 \times 16 = 480.

  4. Bacteria at the end of the n-th hour: 30×230 \times 2ⁿ.

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