对于问题“若ａ,ｂ为正数 ,并且ａ+ｂ =1 ,则有不等式ａ2 +1 +ｂ2 +1≥ 5 .”文 [1 ]给出了较为复杂的代数证法 .之后 ,文 [2 ]给出了简明的几何证法 ,并进行了如下推广 :定理 1　若ａ1 ,ａ2 ,… ,ａｎ ∈Ｒ+,且∑ｎｉ=1ａｉ =1 ,则 ∑ｎｉ =1ａ2ｉ +1≥ｎ2 +1 .文 [2 ]对定理 For the question “If a, b is positive, and a+b =1, then there is an inequality a2 +1 +b2 +1 ≥ 5 .” [1] gives a more complex algebraic proof method. After that, the text [ 2] gives a concise geometric proof method, and carries out the following generalization: Theorem 1 If a1, a2, ..., an ∈ R+, and ∑ni=1ai =1, then ni =1a2i +1≥n2 +1 .[2] Pair Theorem