Question 1171708: The positive integers are arranged in the pattern illustrated below. If this pattern continues indefinitely, what is the number immediately above 39863
A) 39464
B) 39861
C) 39466
D) 39468
E) 39467
https://ibb.co/cx5XHph
Found 2 solutions by mccravyedwin, Edwin McCravy:
Answer by mccravyedwin(408)   (Show Source):
You can put this solution on YOUR website!
row
1 1 2 5 10 17 26 ...
2 3 4 7 12 19 28 ...
3 6 8 9 14 21 30 ...
4 11 13 15 16 23 32 ...
5 18 20 22 24 25 34 ...
6 27 29 31 33 35 36 ...
... ... ...
The diagonal elements are the squares, 1,4,9,16,25,... and the numbers right
after the squares are 2,7,14,23,34, which means we have added 1,3,5,7,9... to
each square to get the next term. So the next term after the square n² is
n² + the nth odd integer. The nth odd integer is 2n-1.
Let's see if that's enough information to determine whether 39863 is a number
to the right of a perfect square diagonal element or to the left of a perfect
square diagonal element.
The square root of 39863 is 199.6572062. So either
39863 is on the 199th row to the right of diagonal element 199²=39601
OR
39863 is on the 200th row to the left of the diagonal element 200²=40000.
The term just right of 199² is 199² plus the 199th positive odd number, which
is 2(199)-1=397. But when we add 397 to the diagonal element 199²=39601, we
get 39998 which is more than 39863, so 39863 cannot be to the right of 39601,
so 39863 must be on the 200th row to the left of the diagonal element
200²=40000.
The number just left of each square on the diagonal is 1 less than the square
on the diagonal.
Now let's observe the numbers left of the diagonal.
Let's look at the numbers left of the diagonal elements beginning with the 2nd
row, as there are no numbers left of the 1 on the first row. They are
3,8,15,24,..., which are 1 less than the squares on the diagonal. We observe
they are the last terms of an arithmetic sequence with common difference 2.
So the 200th term on the 200th row is 40000
The 199th term is 1 less or 39999.
The common ratio is 2.
 <--- the first number on the 200th row
Now we find the number of term on the 200th row which 39863 is.
So 39863 is the 131st term on the 200th row.
So we must now find the number immediately above that, which is
the 131st term on the 199th row.
the 199th term of the 199th row is 39601
The 198th term is 1 less or 39600.
The common ratio is 2.
 <--- the first number on the 199th row
Now we find the 131st term on the 199th row
So 39466 is the 131st term on the 199th row, which is the term immediately
above 39863.
Answer = 39466
Edwin
Answer by Edwin McCravy(20060)  (Show Source):
You can put this solution on YOUR website!
row
1 1 2 5 10 17 26 ...
2 3 4 7 12 19 28 ...
3 6 8 9 14 21 30 ...
4 11 13 15 16 23 32 ...
5 18 20 22 24 25 34 ...
6 27 29 31 33 35 36 ...
... ... ...
The diagonal elements are the squares, 1,4,9,16,25,... and the numbers right
after the squares are 2,7,14,23,34, which means we have added 1,3,5,7,9... to
each square to get the next term. So the next term after the square n² is
n² + the nth odd integer. The nth odd integer is 2n-1.
Let's see if that's enough information to determine whether 39863 is a number
to the right of a perfect square diagonal element or to the left of a perfect
square diagonal element.
The square root of 39863 is 199.6572062. So either
39863 is on the 199th row to the right of diagonal element 199²=39601
OR
39863 is on the 200th row to the left of the diagonal element 200²=40000.
The term just right of 199² is 199² plus the 199th positive odd number, which
is 2(199)-1=397. But when we add 397 to the diagonal element 199²=39601, we
get 39998 which is more than 39863, so 39863 cannot be to the right of 39601,
so 39863 must be on the 200th row to the left of the diagonal element
200²=40000.
The number just left of each square on the diagonal is 1 less than the square
on the diagonal.
Now let's observe the numbers left of the diagonal.
Let's look at the numbers left of the diagonal elements beginning with the 2nd
row, as there are no numbers left of the 1 on the first row. They are
3,8,15,24,..., which are 1 less than the squares on the diagonal. We observe
they are the last terms of an arithmetic sequence with common difference 2.
So the 200th term on the 200th row is 40000
The 199th term is 1 less or 39999.
The common ratio is 2.
<--- the first number on the 200th row
Now we find the number of term on the 200th row which 39863 is.
So 39863 is the 131st term on the 200th row.
So we must now find the number immediately above that, which is
the 131st term on the 199th row.
the 199th term of the 199th row is 39601
The 198th term is 1 less or 39600.
The common ratio is 2.
<--- the first number on the 199th row
Now we find the 131st term on the 199th row
So 39466 is the 131st term on the 199th row, which is the term immediately
above 39863.
Answer = 39466
Edwin
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