The positive integers are arranged in a continuation of the pattern shown below. The value of the integer in the 72nd row (horizontal) and the 15th column (vertical) is
A) 2402 B) 3670 C) 3688 D) 3472 E) 2416
Pattern:
1 3 4 10 11
2 5 9 12
6 8 13
7 14
15
Hey, oobleck
I don't understand your explanation can you elaborate?
734798432789342
Well, looks like we have a number pattern here. But first, let's give the numbers some space to breathe, shall we?
1 3 4 10 11
2 5 9 12
6 8 13
7 14
15
Now, let's try to find a rule here. It seems like the numbers are increasing diagonally from left to right, starting from the top left corner.
Let's look at the diagonal arrows:
1 3 4 10 11
2 5 9 12
6 8 13
7 14
15
Notice how the first diagonal arrow goes from 1 to 3 to 10 to 11. The second arrow goes from 2 to 5 to 9 to 12. And so on.
So, in order to find the value in the 72nd row and 15th column, we need to find the number that would be at the end of the 72nd diagonal arrow.
To do that, let's look at some patterns in the diagonal arrows:
1 - 3 (+2)
3 - 10 (+7)
10 - 21 (+11)
21 - 36 (+15)
...
It seems like the difference between consecutive numbers in the diagonal arrows is increasing by 5 each time. So, let's use this pattern to find the value in the 72nd row.
First, we find the difference between the numbers at the beginning and end of the diagonal arrow in the first row:
11 - 1 = 10
Then, we add this difference (10) to the next number in the diagonal arrow:
11 + 10 = 21
We repeat this process with the new difference (15) and the next number in the diagonal arrow:
21 + 15 = 36
And so on... until we reach the 72nd row:
36 + (5 * (72-1)) = 36 + (5 * 71) = 36 + 355 = 391
So, the value of the integer in the 72nd row and 15th column is 391.
But wait, that's not one of the answer choices! Uh-oh, looks like someone played a joke on us.
In that case, I guess I have to be the clown and tell you that the answer is none of the above.
Hey, sometimes life is just full of unexpected surprises, right?
To find the value of the integer in the 72nd row and 15th column, we can analyze the given pattern and identify any rules or sequences.
Looking at the pattern, we notice that the integers are arranged diagonally in a triangular shape, starting from the top left corner. We can break down the pattern into separate diagonals:
1st diagonal: 1
2nd diagonal: 3, 2
3rd diagonal: 4, 5, 6
4th diagonal: 10, 9, 8, 7
5th diagonal: 11, 12, 13, 14, 15
From observing the pattern, we can see that each diagonal starts with the smallest number in that diagonal and increments by 1 for each subsequent number. In other words, the nth diagonal starts with the number n and ends with the number (n + (n - 1)).
Now, let's determine the diagonal where the element (integer) at the 72nd row and 15th column is located.
To find the diagonal number, we can sum the row number and the column number: 72 + 15 = 87.
The diagonal where the 72nd row and 15th column is located would be the 87th diagonal.
We can now find the range of values within the 87th diagonal:
Starting number of the 87th diagonal = 87
Ending number of the 87th diagonal = 87 + (87 - 1) = 173
Since the 87th diagonal begins at 87 and ends at 173, we know that the integer at the 72nd row and 15th column would be the 72nd element in that diagonal.
To find the 72nd element in the 87th diagonal, let's determine the number of elements in the diagonal preceding the 87th diagonal:
Total elements in diagonals 1 to 86 = 1 + 2 + 3 + ... + 86 = 86 x (86 + 1) / 2 = 3716
Subtracting the total elements in the preceding diagonals (3716) from the 72nd row (72), we will obtain the position of the 72nd element in the 87th diagonal:
Position of 72nd element in 87th diagonal = 72 - 3716 = -3644
Since the position is negative, it means that the 72nd element in the 87th diagonal is actually counted in the opposite direction, starting from the last element of the diagonal.
Therefore, the value of the integer in the 72nd row and 15th column is the 3644th (from the last) element in the 87th diagonal.
The value of the integer can be determined using the equation:
Value = Ending number of the diagonal - Position of the element
Value = 173 - 3644 = -3471
However, since we are looking for positive integers, we can ignore the negative sign and take the absolute value:
Value = | -3471 | = 3471
Therefore, the value of the integer in the 72nd row and 15th column is 3471.
Therefore, the answer choice is not among the options provided (A) 2402, (B) 3670, (C) 3688, (D) 3472, (E) 2416.
first, note that the element at (72,15) is the 15th element along the diagonal starting at row 86, which starts at (86,1) and moves up to the right.
the even-numbered rows start with 1 + the previous odd-numbered row
The odd-numbered row n starts with the value n(n+1)/2
so the element at (86,1) is 1 + 85*86/2 = 3656
so the element at (72,15) is 3656 + 14 = 3670