S1 = 1 cm (initial size)
S2 = S1 + 1/3 S1 = 4/3 cm
S3 = S2 + 1/3 S2 = 7/3 cm
S4 = S3 + 1/3 S3 = 10/3 cm
...
We can observe a pattern here. The size of the shell at each stage is obtained by adding 1/3 of the previous stage's size.
In general, the size of the shell at the nth stage can be expressed as:
Sn = S1 + (1/3)S1 + (1/3)^2S1 + ... + (1/3)^(n-1)S1
To determine the number of shells the crab sheds (excluding the initial shell), we need to find the value of n for which Sn ≥ 10 cm.
10 cm ≤ Sn
10 cm ≤ S1 + (1/3)S1 + (1/3)^2S1 + ... + (1/3)^(n-1)S1
10 cm ≤ S1[1 + (1/3) + (1/3)^2 + ... + (1/3)^(n-1)]
We can recognize the expression inside the brackets as the sum of a geometric series with the first term 1 and the common ratio 1/3. The sum of a geometric series is given by:
Sum = a1 / (1 - r), where a1 is the first term and r is the common ratio.
Plugging in a1 = 1 and r = 1/3, we get:
Sum = 1 / (1 - 1/3) = 3/2
Therefore,
10 cm ≤ S1[3/2]
S1 ≥ (10 cm) * (2/3)
S1 ≥ 6.67 cm
This implies that the crab will shed its shell when it reaches a size of approximately 6.67 cm.
To determine the number of shells shed, we need to find the value of n such that Sn ≥ 6.67 cm.
6.67 cm ≤ S1 + (1/3)S1 + (1/3)^2S1 + ... + (1/3)^(n-1)S1
6.67 cm ≤ S1[1 + (1/3) + (1/3)^2 + ... + (1/3)^(n-1)]
Since the sum of the geometric series is 3/2, we have:
6.67 cm ≤ S1 * (3/2)
S1 ≥ (6.67 cm) * (2/3)
S1 ≥ 4.45 cm
This means that the crab will shed its shell once its size exceeds 4.45 cm.
Now we need to determine the number of stages or molts the crab undergoes before its size exceeds 4.45 cm.
Starting with S1 = 1 cm, we can calculate the subsequent stages as follows:
S2 = 1 cm + (1/3) cm = 4/3 cm
S3 = 4/3 cm + (1/3) * 4/3 cm = 7/3 cm
S4 = 7/3 cm + (1/3) * 7/3 cm = 10/3 cm
We can see that S4 is greater than 4.45 cm. Therefore, the crab will shed its shell during the transition from the third stage (S3) to the fourth stage (S4).
So, the crab will have shed 3 shells (excluding the initial shell) by the time its size exceeds 10 cm.