A plant grows 1.67 cm in its first week. Each week it grows by 2% more than it did the week before. By how much does it grow in twenty weeks, including the first week?
The Correct Answer and Explanation is:
To solve this problem, we need to calculate the growth of the plant over the course of 20 weeks, where the growth increases by 2% each week.
Step-by-Step Calculation:
- First Week Growth: The initial growth is given as 1.67 cm in the first week.
- Growth Pattern: Every week, the growth increases by 2%. This means the growth in the second week will be 1.02 times the growth of the first week, the growth in the third week will be 1.02 times the growth of the second week, and so on.
- Geometric Sequence: The plant’s growth follows a geometric progression, where:
- The first term a1=1.67 cma_1 = 1.67 \, \text{cm}a1=1.67cm.
- The common ratio r=1.02r = 1.02r=1.02 (since the growth increases by 2% each week).
- The number of terms is 20 (weeks).
The formula for the sum SnS_nSn of the first nnn terms of a geometric series is:Sn=a1×1−rn1−rS_n = a_1 \times \frac{1 – r^n}{1 – r}Sn=a1×1−r1−rn
Where:
- a1=1.67a_1 = 1.67a1=1.67
- r=1.02r = 1.02r=1.02
- n=20n = 20n=20
- Substituting the values:
S20=1.67×1−1.02201−1.02S_{20} = 1.67 \times \frac{1 – 1.02^{20}}{1 – 1.02}S20=1.67×1−1.021−1.0220
Now, we calculate 1.02201.02^{20}1.0220:1.0220≈1.48591.02^{20} \approx 1.48591.0220≈1.4859
Substitute this back into the equation:S20=1.67×1−1.4859−0.02S_{20} = 1.67 \times \frac{1 – 1.4859}{-0.02}S20=1.67×−0.021−1.4859S20=1.67×−0.4859−0.02S_{20} = 1.67 \times \frac{-0.4859}{-0.02}S20=1.67×−0.02−0.4859S20=1.67×24.295S_{20} = 1.67 \times 24.295S20=1.67×24.295S20≈40.6 cmS_{20} \approx 40.6 \, \text{cm}S20≈40.6cm
Final Answer:
The plant grows approximately 40.6 cm in 20 weeks, including the first week.
