the positive integers are arranged in the pattern illustrated below if this pattern continues indefinitely, what is the number immediately above 39863?
1 2 5 10 17 26
3 4 7 12 19 28
6 8 9 14 21 30
11 13 15 16 23 32
18 20 22 24 25 34
27 29 31 33 35 36
a) 39464 b)39861 c)39466 d)39468 e)39467
+26367 the positive integers are arranged in the pattern illustrated below if this pattern continues indefinitely, what is the number immediately above 39863?
1 2 5 10 17 26
3 4 7 12 19 28
6 8 9 14 21 30
11 13 15 16 23 32
18 20 22 24 25 34
27 29 31 33 35 36
Let row = n
Let column = k
Formula:
\(\begin{array}{|rcll|} \hline T(n,k) &=& (k-1)^2+2n-1 \qquad & n\le k \\ T(n,k)&=&(n-1)^2+2k \qquad & n>k \\ \hline \end{array}\)
Source: http://oeis.org/search?q=a185725&sort=&language=english
1. \(\begin{array}{|rcll|} \hline \sqrt{39863} &=& 199.657\ldots \\ n_{min} &=& 199 \\ n_{max} &=& 200 \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline T(200,k) &=& (200-1)^2+2k \qquad & n>k \\ 39863 &=& 199^2+2k \\ 2k &=& 39863 - 199^2 \\ 2k &=& 262 \\ \mathbf{k}& \mathbf{=}& \mathbf{131} \\ \hline \end{array} \)
The number immediately above:
\(\begin{array}{|rcll|} \hline T(199,131) &=& (199-1)^2+2\cdot 131 \qquad & n>k \\ T(199,131) &=& 198^2+2\cdot 131 \\ \mathbf{T(199,131)} & \mathbf{=}& \mathbf{39466} \\ \hline \end{array}\)
+773 This pattern is a little difficult to understand because it is difficult to find a pattern.
We can restate the pattern to make it easier for us:
1 2 5 10 17 26
3 4 7 12 18 28
6 8 9 14 21 30
11 13 15 16 23 32
18 20 22 24 25 34
27 29 31 33 35 36
On the first row, I see an additional change of 1, 3, 5, 7, and 9.
The second row has an additional change of 1, 3, 5, 6, and 10.
If we check the first column, the additional change is 2, 3, 5, 7, and 9.
Notice that the digit in the last column increases by 2 every column.
The one directly above 39863 is C) 39466.
- PM
+26367 the positive integers are arranged in the pattern illustrated below if this pattern continues indefinitely, what is the number immediately above 39863?
1 2 5 10 17 26
3 4 7 12 19 28
6 8 9 14 21 30
11 13 15 16 23 32
18 20 22 24 25 34
27 29 31 33 35 36
Let row = n
Let column = k
Formula:
\(\begin{array}{|rcll|} \hline T(n,k) &=& (k-1)^2+2n-1 \qquad & n\le k \\ T(n,k)&=&(n-1)^2+2k \qquad & n>k \\ \hline \end{array}\)
Source: http://oeis.org/search?q=a185725&sort=&language=english
1. \(\begin{array}{|rcll|} \hline \sqrt{39863} &=& 199.657\ldots \\ n_{min} &=& 199 \\ n_{max} &=& 200 \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline T(200,k) &=& (200-1)^2+2k \qquad & n>k \\ 39863 &=& 199^2+2k \\ 2k &=& 39863 - 199^2 \\ 2k &=& 262 \\ \mathbf{k}& \mathbf{=}& \mathbf{131} \\ \hline \end{array} \)
The number immediately above:
\(\begin{array}{|rcll|} \hline T(199,131) &=& (199-1)^2+2\cdot 131 \qquad & n>k \\ T(199,131) &=& 198^2+2\cdot 131 \\ \mathbf{T(199,131)} & \mathbf{=}& \mathbf{39466} \\ \hline \end{array}\)
