Starting from the point (0,0), a grasshopper makes a series of leaps on the coordinate plane. The grasshopper's first leap takes her to (3,4). After a move which adds (x,y) to the grasshopper's coordinates, her next move adds either (x,y-1) or (x-1,y) to her coordinates. Thus her second move adds either (3,3) or (2,4) to her position, landing her at (6,7) or (5,8) accordingly.
If the grasshopper's x and y coordinates never decrease, and if she ends at the point (a,a) where a is as large as possible, then what is the value of a?
Give an explanation with the answer. Thanks!!!! :)
14
14
nitially, $s = 0$. After the first move, the grasshopper is at $(3,4)$, so $s = 7$. The next move adds either $(3,3)$ or $(2,4)$. In either case, $s$ must increase by $6$, up to $13$. In the next moves, $s$ must increase by $5$, then $4$, then $3$, then $2$, then $1$. Finally, the grasshopper stops because she cannot continue moving without decreasing at least one of her coordinates.
After these moves, $s$ is equal to $7 + 6 + 5 + 4 + 3 + 2 + 1 = 28$. Since the trip ends at the point $(a,a)$, we have $28= s = 2a$. Therefore, $a = \boxed{14}$.
Notice that it is possible to get to $(14,14)$ by following the path: $(0,0)\rightarrow (3,4) \rightarrow (6,7) \rightarrow (9,9) \rightarrow (11,11) \rightarrow (13,12) \rightarrow (14,13) \rightarrow (14,14)$.
Well, let's trace the grasshopper's movements step by step. The first leap takes her to (3,4).
For the second move, she can either add (3,3) or (2,4) to her position. If she adds (3,3), she will be at (3 + 3, 4 + 3) = (6,7). If she adds (2,4), she will be at (3 + 2, 4 + 4) = (5,8).
Now, let's analyze the possible moves for the third step. If the grasshopper is at (6,7), her options are to add either (6,6) or (5,7). If she adds (6,6), she will be at (6 + 6, 7 + 6) = (12,13). If she adds (5,7), she will be at (6 + 5, 7 + 7) = (11,14).
Continuing this pattern, we can see that each time the grasshopper has two options for her next move. One option will increase both x and y coordinates by the same amount, while the other option will increase the x coordinate by 1 and the y coordinate by 1 less than the x coordinate.
We can observe that the grasshopper always has two options for her next move, but the sum of the coordinates always increases by 1. Therefore, when the grasshopper ends at the point (a,a), the value of a will be equal to the total number of steps she took.
Let's count the steps she took:
- First leap: (0 --> 3, 0 --> 4)
- Second leap: (3 --> 6, 4 --> 7) or (3 --> 5, 4 --> 8)
- Third leap: (6 or 5 --> 12 or 11, 7 or 8 --> 13 or 14)
From this pattern, we can see that for each move, the sum of the coordinates increases by 1. Thus, the total number of steps she took will be equal to the sum of the coordinates of her final position.
In this case, her final position is (a,a), so the value of a is equal to the total number of steps she took. Each step increases the sum of the coordinates by 1, so a is equal to the sum of all the coordinates she visited.
Let's calculate the sum of the coordinates:
0 + 3 + 0 + 4 + 3 + 4 + 6 + 7 + 12 + 13 + ...
Notice that the sum of the first two coordinates, (0 + 3 + 0 + 4), is equal to the sum of the third leap's coordinates, (6 or 5 + 7 or 8). This pattern continues throughout all the steps.
So, the sum of the coordinates can be simplified as follows:
(0 + 3 + 0 + 4) + (6 or 5 + 7 or 8) + (12 or 11 + 13 or 14) + ...
The sum of the first two coordinates is 7, which is equal to the sum of the next leap's coordinates. And the sum of the next leap's coordinates is equal to the sum of the coordinates of the following leap.
Therefore, the sum of the coordinates becomes an infinite geometric series with a common ratio of 1 and a first term of 7.
Using the formula for the sum of an infinite geometric series, the sum of the coordinates can be calculated as follows:
Sum = a / (1 - r)
Where a is the first term (7) and r is the common ratio (1).
Using this formula, the sum of the coordinates is calculated as:
Sum = 7 / (1 - 1) = 7 / 0 (undefined)
Since the sum is undefined, it means that the grasshopper will never reach a point where the x and y coordinates are equal. This implies that there is no value of a that satisfies the given conditions.
So, the value of a is undefined in this case.
To find the value of a, we need to figure out the Grasshopper's movement pattern and observe its behavior.
Let's start by identifying a pattern in the grasshopper's movements:
1. The initial position is (0,0)
2. The first move is (3,4)
3. The next move can be either (3,3) or (2,4)
4. Subsequent moves can be determined based on the previous move, following the condition that the x and y coordinates never decrease.
We can create a table to determine the possible moves of the grasshopper:
| Move | (x, y) | Next Move |
|------|--------|-----------|
| 1 | (0,0) | (3,4) |
| 2a | (3,4) | (6,7) |
| 3a | (6,7) | (9,11) |
| 4a | (9,11) | (12,16) |
Alternately,
| Move | (x, y) | Next Move |
|------|--------|-----------|
| 1 | (0,0) | (3,4) |
| 2b | (3,4) | (5,8) |
| 3b | (5,8) | (7,13) |
| 4b | (7,13) | (9,19) |
By observing these moves, we can notice that there are two different patterns (2a, 3a, 4a) and (2b, 3b, 4b) depending on the second move:
In pattern (2a, 3a, 4a), the x-coordinate increases by 3, and the y-coordinate increases by 4 in each move.
In pattern (2b, 3b, 4b), the x-coordinate increases by 2, and the y-coordinate increases by 4 in each move.
Since we want to find a point (a, a) where a is as large as possible, we need to identify the pattern that would lead to the largest possible value for a.
In pattern (2a, 3a, 4a), the x-coordinate increases by 3 units, whereas in pattern (2b, 3b, 4b), the x-coordinate only increases by 2 units. Consequently, pattern (2a, 3a, 4a) will result in a larger value for a.
Hence, we need to determine how many times we can apply pattern (2a, 3a, 4a) before the grasshopper's coordinate decreases.
Looking at the pattern (2a, 3a, 4a), the x-coordinate increases by 3, and the y-coordinate increases by 4. So, in the n-th move for this pattern, the grasshopper's position will be (3n, 4n).
To find the n-th move where the x coordinate increases and doesn't decrease, we need to set 3n greater than the previous x coordinate:
3n > (3n-3)
Simplifying the inequality:
3n > 3n - 3
The inequality states that 3n is greater than 3n - 3. This is always true regardless of the value of n since any number is always greater than itself minus 3.
Therefore, there is no restriction on n, which means the x-coordinate will never decrease in the pattern (2a, 3a, 4a).
Now, to find the largest possible value for a, we need to find the value of a when n is maximum. From the last equation, we can see that n can be any positive integer, so let's consider n = 1, which yields:
x = 3n = 3(1) = 3
y = 4n = 4(1) = 4
Therefore, the Grasshopper's final position is (3, 4) when n = 1, and a = max(x, y) = max(3, 4) = 4.
Hence, the value of a is 4.