Question

小学奥数裂项计算，高手请进！

1/(3²-1)+1/(5²-1)+1/(7²-1)+……+1/(13²-1)=

Answer 1

1/(3²-1)+1/(5²-1)+1/(7²-1)+……+1/(13²-1)
=1/[(3-1)(3+1)]+1/[(5-1)(5+1)]+1/{(7-1)(7+1)]+……+1/[(13-1)(13+1)]
=1/(2*4)+1/(4*6)+1/(6*8)+......+1/(12*14)

=[2/(2*4)+2/(4*6)+2/(6*8)+......+2/(12*14)]/2
=[(1/2-1/4)+(1/4-1/6)+(1/6-1/8)+......(1/12-1/14)]/2
=(1/2-1/4+1/4-1/6+1/6-1/8+......1/12-1/14)/2
=(1/2-1/14)/2
=(7/14-1/14)/2
=(6/14)/2
=(3/7)/2
=3/14

Answer 2

原试=1/(4×2）+1/(6×4）+1/(8×6）···+1/(14×12）
=1/2（1/2-1/4)+1/2(1/4-1/6)....+1/2(1/12-1/14)
=1/2(1/2-1/4+1/4-1/6+....+1/12-1/14)
=1/2（1/2-1/14)
=1/2×3/7
=1/14

Answer 3

原式 = 1/(3-1)(3+1) + 1/(5-1)(5+1)+....+1/(13-1)(13+1)
= 1/(2x4) + 1/(4x6) + .... + 1/(12x14)
= [1/2-1/4 + 1/4-1/6 + ... + 1/12-1/14]/2
=[1/2 - 1/14]/2 = 3/14

Answer 4

1/(3²-1)={1/(3-1)-1/(3+1)}/2

以此类推
原式={1/(3-1)-1/(13+1)}/2=3/14

Answer 5