數學問題.1+2+3+......+n的和等於一個三位數
n為正整數,若1+2+3+......+n的和等於一個三位數,且此三位數的每個數字皆相同(如:111,222,333,444......).試找出所有可能的n
n=36 1+2+3+......+36=0.5x(36) x(36+1)= 666 1+2+3+......+n= (1/2)x n x (n+1) if (1/2)x n x (n+1)=111 then n(n+1)=222 =2 x 111=2x3x37=6x37, thus n does not exist. if (1/2)x n x (n+1)=222 then n(n+1)=444=4x111=12x37, thus n does not exist. 111=3x37 333x2=666=18x37, n does not exist. 444x2=888=24x37, n does not exist. 555x2=1110=30x37, n does not exist. 666x2=1332=36x37, Yeah 777x2=1554=42x37, n does not exist. Rests are same So that, n=36 and 1+2+3+......+36=0.5x(36) x(36+1)= 666
n=6因為1+2+3+......+n=0.5*n*(n+1)所以0.5*n*(n+1)=111,222,333,444,555,666,777,888,999所以n*(n+1)=222,444,666,888,1110,1332,1554,1776,199814*15<222<15*1620*21<444<21*2224*25<666<25*2629*30<888<30*3132*33<1110<33*3435*36<1332=36*3738*39<1554<39*4041*42<1776<42*4344*45<1998<45*46所以只有36*37=1332所以n=36參考資料：我(1+n)*n/2=666n^2+n-1332=0(n+37)(n-36)=0n=-37ORn=36n=-37不成立故n=36
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