你要求和的数列是1,﹣2ˆ2,3ˆ2,﹣4ˆ2,5ˆ2,﹣6ˆ2,······,(-1)ˆ(n-1)×nˆ2吧.
当n为偶数时,Sn=1ˆ2﹣2ˆ2＋3ˆ2﹣4ˆ2＋5ˆ2﹣6ˆ2＋···＋(n-1)ˆ2﹣nˆ2
=﹣[(2ˆ2﹣1ˆ2)＋(4ˆ2﹣3ˆ2)＋(6ˆ2﹣5ˆ2)＋···＋((n-1)ˆ2﹣nˆ2)
=﹣[(2+1)＋(4+3)＋(6+5)＋···＋(2n-1)]
=﹣[3＋7＋11＋···＋(2n-1)]
=﹣n[3＋(2n-1)]/2
=﹣nˆ2/2
当n为奇数时,n-1为偶数,Sn=S(n-1)＋n=﹣(n-1)ˆ2/2＋n =﹣(nˆ2+1)/2