1×2+2×3+3×4+……+48×49+49×50等于多少?希望有解答的具体步骤!谢谢!

2024-12-16 08:14:30
推荐回答(2个)
回答1:

通项式是an=n(n+1)=n²+n
所以Sn=n(n+1)(2n+1)/6+n(n+1)/2
[n(n+1)(2n+1)/6是平方和求和公式,没学过的话要用数学归纳法证明]
所以S50=50×51×101/6+50×51/2=44,200

回答2:

利用等式:
n(n+1) = (1/3)[(n+2)(n+1)(n) - (n+1)(n)(n-1)]

1×2+2×3+3×4+……+48×49+49×50
=(1/3)[(3×2×1 - 0) + (4×3×2 - 3×2×1) + ... + (51×50×49 - 50×49×48)]
=(1/3)(51×50×49)(上步中前后抵消)
=41650