Final answer:
By using a geometric progression, we can find that the total number of leaves that have fallen to the ground by the end of the 24th day of autumn is 116,226 leaves.
Step-by-step explanation:
To find the total number of leaves that have fallen to the ground by the end of the 24th day of autumn, we need to determine the pattern of leaf falling. Based on the information given, the number of leaves on the ground is tripling with each passing day. So, we can create a sequence to represent the number of leaves falling each day.
Starting from the 5th day, we can write the sequence:
- Day 5: 1 leaf
- Day 6: 3 leaves (1 leaf tripled)
- Day 7: 9 leaves (3 leaves tripled)
- Day 8: 27 leaves (9 leaves tripled)
- ...
- Day 24: ?
To find the number of leaves on the 24th day, we can use the formula for finding the nth term of a geometric progression:
an = a1 * r(n-1)
where a1 is the first term, r is the common ratio, and n is the term number.
In this case, a1 = 1 (number of leaves on the 5th day) and r = 3 (ratio of leaf falling each day).
Plugging in the values, we can find the number of leaves on the 24th day:
a24 = 1 * 3(24-5)
a24 = 1 * 319
a24 = 1 * 116,226
a24 = 116,226
Therefore, the total number of leaves that have fallen to the ground by the end of the 24th day of autumn is 116,226 leaves.