Generalized Formula For Consecutive Square Number’sArithmetic Progression in Hyper Dimension or Multi Dimension [Hyper Dimension or Multi Dimension = 2 Dimension, 3 Dimension, 4 Dimension…….Nth Dimension or infinite Dimension]

1. Generalized Formula For
Consecutive Square Number’s
Arithmetic Progression in Hyper
Dimension or Multi Dimension
[Hyper Dimension or Multi Dimension
= 2 Dimension, 3 Dimension, 4
Dimension…….Nth Dimension or infinite
Dimension]

2. Formula
A.P [(N-D)^2]
= H(N) * [ F+L ] + [ μ + H(N)*[ F * (F+(H(N)-3 ) ) ]]

3. Formula Symbol meanings
H(N) = R * C * Face * Cube * …. * Nth
Dimension
F = First Number in square
L = Last Number in square
μ = Hyper dimensional Constant

4. Hyper dimensional Constant
1. If H(N) greater than or equal to 5
then;
퐻(푁)[ 퐴 ∗ 퐴 − 4 + 1]
μ = 2[ H(N) -3 ] + 퐴=5
2. If H(N) from 1 to 4
then ; {H(N), μ} => { 1, 0} , {2,-1},{3,-1},{4,2}

5. Example : 3 Dimension
A.P [(N-D)^2]
= H(N) * [ F+L ] + [ μ + H(N)*[ F * (F+(H(N)-3 ) ) ]]
A.P [(3-D)^2]
= [R*C*Face] * [ F+L ] + [ μ + [R*C*Face] *[ F *
(F+([R*C *Face] -3 ) ) ]]

6. Example 3 Dimension
3 D
6^2 7^2
3^2
8^2 9^2
2^2
4^2 5^2

7. Example 3 Dimension with calculations
A.P [(3-D)^2]
= [R*C*Face] * [ F+L ] + [ μ + [R*C*Face] *[ F * (F+([R*C *Face] -3 ) ) ]]
Here R=2, C= 2, Face =2 , F= 2, L =9
so H(N) = R*C*Face = 2* 2 * 2
μ condition, H(N) = 8, H(N)>= 5
퐻(푁)[ 퐴 ∗ 퐴 − 4 + 1]
μ = 2[ H(N) -3 ] + 퐴=5
8 [ 퐴 ∗ 퐴 − 4 + 1]
= 2[ 8 -3 ] + 퐴=5
= 2[5] + [5+1] +[12+1]+[21+1]+[33+1]
=10+6+13+22+33
= 84

8. Example 3 Dimension with calculations
A.P [(3-D)^2]
= [R*C*Face] * [ F+L ] + [ μ + [R*C*Face] *[ F * (F+([R*C
*Face] -3 ) ) ]]
= [8] *[2+9] + [84 + [8*[2* 2+(8-3)]]]
= 88+[84+8*[14]]
= 88 + 84+ 112 = 88 + 196
= 284

9. Checking
2^2+ 3 ^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2
= 4+9+16+25+36+49+64+81
=284
 Finally …. Happy