第一題,
13又1/6+15又1/12+17又1/20+19又1/30
= 13+1/6 + 15+1/12 + 17+1/20 + 19+1/30
= (13+15+17+19) + 1/6+1/12 +1/20+1/30
= 64 + 1/(2*3)+1/(3*4) +1/(4*5)+1/(5*6)
= 64 + 1/2-1/3 + 1/3-1/4 + 1/4-1/5 + 1/5-1/6
= 64 + 1/2 + 1/6
= 64 + 2/6
= 64 + 1/3
= 64又1/3
第二題,
1/1x4+1/4x7+1/7x10+...+1/97x100
= 1/3 * { 3/(1x4) + 3/(4x7) + 3/(7x10) + ...+ 3/(97x100) }
= 1/3 * { (1/1-1/4) + (1/4-1/7) + (1/7-1/10) + ...+ (1/97-1/100) }
= 1/3 * (1/1-1/100)
= 33/100
第三題,
2/3+2/15+2/35+2/63+2/99+2/143
= 2/(1x3)+2/(3x5)+2/(5x7)+2/(7x9)+2/(9x11)+2/(11x13)
= 1/1-1/3 + 1/3-1/5 + 1/5-1/7 + 1/7-1/9 + 1/9-1/11 + 1/11-1/13
= 1 - 1/13
= 12/13
第四題,
1又1/10+3又1/40+5又1/88+7又1/154+9又1/238+11又1/340
= 1+1/10+3+1/40+5+1/88+7+1/154+9+1/238+11+1/340
= (1+3+5+7+9+11) + 1/10+1/40+1/88+1/154+1/238+1/340
= 36 + 1/(2*5)+1/(5*8)+1/(8*11)+1/(11*14)+1/(14*17)+1/(17*20)
= 36 + 1/3 * (1/2-1/5+1/5-1/8+1/8-1/11+1/11-1/14+1/14-1/17+1/17-1/20)
= 36 + 1/3 * (1/2 - 1/20)
= 36 + 1/3 * 9/20
= 36 + 3/20
= 36 又3/20