Question

Lerna's Hydra is a mythological character that appears in some stories such as the 12 Tasks of Hercules. The Hydra was a one-headed monster but when it is cut off, 2 more heads grow in its place. If a hero tried to conquer it by cutting off all of its heads every day, how many heads would the Hydra have on the third day? And at the end of 10 days of trying to kill it?

Answer 1

a. 8 heads. b. 1024 heads.

Explanation:

a. How many heads would the Hydra have on the third day?

Since the Hydra's head doubles for every initial head that is cut off on each day, the number of heads the hydra has after each day is 2ⁿ where n = number of days. Now, the head increase of the Hydra form a geometric sequence with the first term as 2 since we initially had one head and was cut off to produce 2 heads. On the second day, two heads are cut off to produce four which is 2² and so on.

So, on the third day, we have 2³ heads = 8 heads.

b. And at the end of 10 days of trying to kill it?

At the end of 10 days we have n = 10 in 2ⁿ. So, 2¹⁰ = 1024 heads.

So, at the end of 10 days, the Hydra has 1024 heads.