The seed of a particular variety of a rose in a rose garden, multiplies into 2 everyday due to reproduction. If there is only one seed on the first day,it increases to two on the second day,increases to eight on fourth day and so on.If starting with one seed it takes 30 days to cover 200 meters plot,how long will it take to cover the same plot if started with two seeds ?

Solution (By Examveda Team)

The roses are increasing the pattern,
1, 2, 4, 8, 16 and so on in each passing day. The given series is in to Geometric Progression.
Common ratio of this GP,
= ²/₁ = 2
Now, The sum of the given GP, taking one seed first day, i.e
a = 1
Number of terms = 30 (as given number of days are 30.)
Sum = a * [(r^n - 1) /(r -1)]
Sum = 1 * [2^30 - 1)/(2 - 1)]
Sum = 1,07,37,41,823.

1, 07, 37, 41, 823 roses are needed to cover 200 meters plot. Now, in Second case, series would be like this:
2, 4, 8, 16, 32 and so on.
a = 2
r = ⁴/₂ = 2.
Sum = 1, 07, 37, 41, 823.
Now, we have to find number of days, i.e n.
Sum = a * [(r^n - 1) /(r -1)]
1, 07, 37, 41, 823 = 2 *[(2^n - 1) /(2 - 1)]
On Solving you will get,
n = 29 days.

Note:
If you understand then you don't need to do calculation of the bold part. See, On 29^th days number of roses are half as we get in first case that is starting one seed. You can see in the first case
r = 1 and in second case
r = 2. It means it will give double production as in first case in 29 days which is 1, 07, 37, 41, 823 and it will cover 200 meters plot.