A gardener has a field in the shape of an isosceles triangle. He decided to plant it with rose bushes with...?

Question

A gardener has a field in the shape of an isosceles triangle. He decided to plant it with rose bushes with the aid of his apprentice. He planted the first row with 101 bushes along the base, then his apprentice planted the next row with 100 bushes. They continued in this way, planting alternate rows, until the whole field was planted. The gardener planted the last row, which contained 1 bush.
How many more bushes did the gardener plant than the apprentice?

The person to give me the best answer along with a valid reason to back it up, will get 10 points.

A- 100
B- 101
C- 49
D- 50
E- 51

Answer 1

51 because the apprentice always plants 1 less than the gardner after each row and if you count rows 101 to 0 inclusive = 102 rows so half of that is 51 as that is how many rows the gardner planted.

Answer 2

gardener planted the 51 odd rows from 101 to 1

that will be the same as 51 rows of 51 (pair up numbers in order from each end: 101+1, 99+3, 97+5 etc and then halve them)

apprentice planted the 50 even rows from 100 to 2

that will be the same 50 rows of 51

the gardener plants the equivalent of one extra row of 51

Answer 3

Well, we know that the gardener planted all odd number plants. In math form we can say, where G is the number of plants the gardener planted:
G = SUM(i=0 to 50) (2i + 1)

The apprentice planted all even number plants. In math form we can say, where A is the number of plants the apprentice planted:
A = SUM(i=0 to 50) (2i)

Let us find the formula for the sum of odd numbers:
i = 0 SUM = 1
i = 1 SUM = 4
i = 2 SUM = 9
Guess that SUM(i=0 to n) (2i + 1) = (n+1)^2
Prove true for n = 0
SUM(i=0 to 0) (2i + 1) = 2(0) + 1 = 1
(0+1)^2 = 1^2 = 1
Proven for n = 0.
Now if true for n, prove for n +1:
SUM(i=0 to n+1) (2i + 1) = 2(n+1) + 1 + SUM(i=0 to n) (2i + 1)
= 2n + 3 + (n+1)^2
= n^2 +4n + 4
= (n+2)^2
Therefore, by induction we've shown SUM(i=0 to n) (2i +1) = (n+1)^2

Let us find the formula for sum of even numbers:
i=0 SUM = 0
i=1 SUM = 2
i=2 SUM = 6
Guess that SUM(i=0 to n) (2i) = n(n+1)
Prove for n = 0:
SUM(i=0 to 0) (2i) = 2*0 = 0
0(0+1) = 0
Therefore, proven for n = 0.
Now if true for n prove for n + 1:
SUM(i=0 to n+1) (2i) = 2(n+1) + SUM(i=0 to n) (2i)
= 2n + 2 + n(n+1)
= n^2 + 3n + 2
= (n+1)(n+2)
Therefore, by induction we've proven SUM(i=0 to n) (2i) = n(n+1).

Returning to the sums of plants planted by gardener:
G = SUM(i=0 to 50) (2i + 1)
= (50 + 1)^2
= 51^2

Returning to the sums of plants planted by apprentice:
A = SUM(i=0 to 50) (2i)
= 50(50 + 1)
= 50(51)

Finding the difference:
G-A = 51^2 - 51(50)
G-A = 51(51 - 50)
G-A = 51(1)
G-A = 51

Therefore, the gardener planted 51 more plants than the apprentice.