**Question:** In the middle of a round pool lies a beautiful water lily. The water lily doubles in size every day. After exactly 20 days, the lily will cover the complete pool. After how many days will the water lily cover half of the pool?

**Answer:** Because the water lily doubles its size every day and the complete pool is covered after 20 days, half of the pool will be covered one day before that, after 19 days.

Let the size of the lily be

Sin the beginning.At day 2, the size of the lily =

S x 2 = (2^1)(S) = 2(S)

At day 3, the size of the lily =2(S) x 2 = (2^2)(S)= 4(S)

At day 4, the size of the lily =4(S) x 2 = (2^3)(S) = 8(S)

At day 5, the size of the lily =8(S) x 2 = (2^4)(S) = 16(S)

.

.

At day 19, the size of the lily =131072(S) x 2 = (2^18)(S) = 262144(S)

At day 20, the size of the lily =262144(S) x 2 = (2^19)(S) = 524288(S)Therefore, the size of the pool =

(2^19)(S)

Now, the size of half of the pool =

(2^19)(S)/2 = (2^18)(S)

which is the size of the lily at the 19th day.Hence,

After 19 days, the water lily will cover half of the pool.

**Conclusion:** After 19 days, the water lily will cover half of the pool.

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