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Sum of sequence, number of cubes needed to build pyramid

ÀÛ¼ºÀÚ Uploader : fatman ÀÛ¼ºÀÏ Upload Date: 2019-08-16º¯°æÀÏ Update Date: 2019-08-27Á¶È¸¼ö View : 289

When building a pyramid using a cube as shown in the figure, the number N of cubes needed for the number n of layers can be obtained as follows.

If n = 1, a1 = 1
If n = 2, a2 = a1 + 4*1
If n = 3, a3 = a2 + 4*2 = a1 + 4*1 + 4*2
If n = 4, a4 = a3 + 4*3 = a1 + 4*1 + 4*2 + 4*3
       ¡¦
If n = n, an = a1 + 4*(1+2+3+¡¦+(n-1)) = 1 + 4*n*(n-1)/2 = 2n^2 - 2n + 1

N = ¢²ak = ¢²2k^2 - 2k + 1 = 2¢²k^2 - 2¢²k + ¢²1

Summing from k=1 to k=n,

N = n*(n+1)*(2*n+1)/3 - n*(n+1) + n


*** Âü°í¹®Çå[References] ***

N = n*(n+1)*(2*n+1)/3 - n*(n+1) + n
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